autpgrp-1.12.0/0000755000000000000000000000000015202202400010166 5ustar00autpgrp-1.12.0/CHANGES.md0000644000000000000000000000367015202202400011566 0ustar00# CHANGES to the 'autpgrp' GAP package ## 1.12.0 (2026-05-18) - converted the manual to GAPDoc - minor janitorial changes ## 1.11.1 (2025-04-10) - Update an outdated statement in the install instructions (we don't use ".zoo" files anymore) - update Max Horn's contact details yet again ## 1.11 (2022-08-05) - make global functions in this package read-only to prevent accidents caused by overwriting them with different code - update Max Horn's address ## 1.10.1 (2019-07-15) - fixed a manual example and the test derived from it to work with both GAP 4.11 and older GAP versions - made further janitorial changes ## 1.10 (2018-07-30) - added Max Horn as package maintainer - added this file with overview of changes in each package release - moved the internal function `InducedActionFactor` from the GAP library into this package - removed the internal function `BlockPosition` ## 1.9 (2018-03-07) - moved package to GitHub - removed some unused code - added automated test suite ## 1.8 (2016-11-25) - replaced use of obsolete GAP functions `MutableNullMat`, `MutableIdentityMat`, `NormedVectors` by `NullMat`, `IdentityMat` `NormedRowVectors` ## 1.7 (2016-11-20) - changed WedgePlusAction to deal separately with characteristic 2 - added new undocumented functions `NumberOfClass2AssociativeAlgebras` and `NumberOfClass2AssocAlgebrasByDim` - improved orbit calculations to use dictionaries, which drastically improves performance for big orbits (but may slow down very small orbit computations a little bit) - replaced uses of `ConvertToMatrixRep` plus `Immutable` by `ImmutableMatrix` - updated some examples in the package manual; in particular, replaced use of `RequirePackage` by `LoadPackage` - cleaned up some code - stopped using the obsolete GAP variable `Revision` - removed CVS leftovers ## 1.6 (2012-05-29) ## 1.5 (2012-05-29) ## 1.4 (2009-08-31) ## 1.3 (2009-07-03) ## 1.2 (2002-11-19) autpgrp-1.12.0/LICENSE0000644000000000000000000004414215202202400011200 0ustar00The AutPGrp package is free software; you can redistribute and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your opinion) any later version. 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If this is what you want to do, use the GNU Lesser General Public License instead of this License. autpgrp-1.12.0/PackageInfo.g0000644000000000000000000001344615202202400012515 0ustar00############################################################################# ## ## PackageInfo.g for the package `AutPGrp' Bettina Eick ## SetPackageInfo( rec( PackageName := "AutPGrp", Subtitle := "Computing the Automorphism Group of a p-Group", Version := "1.12.0", Date := "17/05/2026", # dd/mm/yyyy format License := "GPL-2.0-or-later", Persons := [ rec( LastName := "Eick", FirstNames := "Bettina", IsAuthor := true, IsMaintainer := true, Email := "beick@tu-bs.de", WWWHome := "http://www.iaa.tu-bs.de/beick", PostalAddress := Concatenation( "Institut Analysis und Algebra\n", "TU Braunschweig\n", "Universitätsplatz 2\n", "D-38106 Braunschweig\n", "Germany" ), Place := "Braunschweig", Institution := "TU Braunschweig"), rec( LastName := "Horn", FirstNames := "Max", IsAuthor := false, IsMaintainer := true, Email := "mhorn@rptu.de", WWWHome := "https://www.quendi.de/math", GitHubUsername := "fingolfin", PostalAddress := Concatenation( "Fachbereich Mathematik\n", "RPTU Kaiserslautern-Landau\n", "Gottlieb-Daimler-Straße 48\n", "67663 Kaiserslautern\n", "Germany" ), Place := "Kaiserslautern, Germany", Institution := "RPTU Kaiserslautern-Landau" ), rec( LastName := "O'Brien", FirstNames := "Eamonn", IsAuthor := true, IsMaintainer := false, Email := "obrien@math.auckland.ac.nz", WWWHome := "http://www.math.auckland.ac.nz/~obrien", PostalAddress := Concatenation( "Department of Mathematics\n", "University of Auckland\n", "Private Bag 92019\n Auckland\n New Zealand\n" ), Place := "Auckland", Institution := "University of Auckland" ), ], Status := "accepted", CommunicatedBy := "Derek F. Holt (Warwick)", AcceptDate := "09/2000", PackageWWWHome := "https://gap-packages.github.io/autpgrp/", README_URL := Concatenation( ~.PackageWWWHome, "README" ), PackageInfoURL := Concatenation( ~.PackageWWWHome, "PackageInfo.g" ), SourceRepository := rec( Type := "git", URL := "https://github.com/gap-packages/autpgrp", ), IssueTrackerURL := Concatenation( ~.SourceRepository.URL, "/issues" ), ArchiveURL := Concatenation( ~.SourceRepository.URL, "/releases/download/v", ~.Version, "/autpgrp-", ~.Version ), ArchiveFormats := ".tar.gz", AbstractHTML := "The AutPGrp package introduces a new function to compute the automorphism group of a finite $p$-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.", PackageDoc := rec( BookName := "AutPGrp", ArchiveURLSubset := ["doc"], HTMLStart := "doc/chap0_mj.html", PDFFile := "doc/manual.pdf", SixFile := "doc/manual.six", LongTitle := "Computing the Automorphism Group of a p-Group", ), Dependencies := rec( GAP := ">=4.7", NeededOtherPackages := [], SuggestedOtherPackages := [], ExternalConditions := [] ), AvailabilityTest := ReturnTrue, TestFile := "tst/testall.g", Keywords := ["p-group", "automorphism group"], AutoDoc := rec( entities := rec( AUTPGRP := "AutPGrp", ), TitlePage := rec( Copyright := """ Bettina Eick and Eamonn O'Brien.

AutPGrp is free software; you can redistribute it and/or modify it under the terms of the https://www.fsf.org/licenses/gpl.html as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.""", Abstract := """ The &AUTPGRP; package introduces a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in &GAP; code and it makes use of a number of methods from the &GAP; library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in &GAP;. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.""", Acknowledgements := """ We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the &GAP; PQuotient which are used in this package. """)), )); autpgrp-1.12.0/README0000644000000000000000000001053415202202400011051 0ustar00 The AutPGrp package ------------------- AutPGrp is a GAP 4 package for computing automorphism groups of p-groups. Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch & Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the GAP 4 library. Recently, Cannon & Holt (1999) presented a new algorithm which uses a ``hybrid group'' approach. More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the GAP 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of p-groups; the GAP 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of GAP 3. Moreover, the p-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite p-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU pq C program. In the AutPGrp package we introduce a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU pq method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. The GAP 4 package ANUPQ, which is an interface to most of the functionality of the ANU pq C program, uses the AutPGrp package to compute automorphism groups of p-groups. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. Note that since version 1.1 of AutPGrp, at least GAP 4.3fix4 is required. Authors ------- The AutPGrp package was written by: Bettina Eick, Institut Computational Mathematics, TU Braunschweig, Pockelsstr. 14, D-38106 Braunschweig, Germany e-mail: beick@tu-bs.de Eamonn O'Brien Department of Mathematics University of Auckland Private Bag 92019, Auckland, New Zealand e-mail: obrien@math.auckland.ac.nz Installing the AutPGrp package ------------------------------ To install the AutPGrp package, move the archive file 'autpgrp.tar.gz' into the `pkg' directory in which you plan to install AutPGrp. Usually, this will be the directory `pkg' in the hierarchy of your version of GAP 4. (However, it is also possible to keep an additional `pkg' directory in your private directories, see section "ref:Installing GAP Packages" of the GAP 4 reference manual for details on how to do this.) Then unpack the archive file and that's it! Bug reports ----------- If you encounter problems, please contact Bettina Eick . When sending a bug report, remember we will need to be able to reproduce the problem; so please include: * The version of GAP you are using; either look at the header when you start up GAP, or at the gap> prompt type: VERSION; * The operating system you are using e.g. Linux, macOS, Windows, ... * A script in GAP that demonstrates the bug, along with a description of why it's a bug (e.g. by adding comments to the script - recall, comments in GAP begin with a #). - Bettina Eick e-mail: b.eick@tu-bs.de www: http://www.iaa.tu-bs.de/beick/so.html autpgrp-1.12.0/doc/0000755000000000000000000000000015202202400010733 5ustar00autpgrp-1.12.0/doc/_AutoDocMainFile.xml0000644000000000000000000000014115202202400014553 0ustar00   autpgrp-1.12.0/doc/_Chunks.xml0000644000000000000000000000000015202202400013035 0ustar00autpgrp-1.12.0/doc/_entities.xml0000644000000000000000000000027215202202400013441 0ustar00AutPGrp'> AutPGrp'> autpgrp-1.12.0/doc/_main.tex0000644000000000000000000010761515202202400012552 0ustar00% generated by GAPDoc2LaTeX from XML source (Frank Luebeck) \documentclass[a4paper,11pt]{report} \usepackage[top=37mm,bottom=37mm,left=27mm,right=27mm]{geometry} \sloppy \pagestyle{myheadings} \usepackage{amssymb} \usepackage[utf8]{inputenc} \usepackage{makeidx} \makeindex \usepackage{color} \definecolor{FireBrick}{rgb}{0.5812,0.0074,0.0083} \definecolor{RoyalBlue}{rgb}{0.0236,0.0894,0.6179} \definecolor{RoyalGreen}{rgb}{0.0236,0.6179,0.0894} \definecolor{RoyalRed}{rgb}{0.6179,0.0236,0.0894} \definecolor{LightBlue}{rgb}{0.8544,0.9511,1.0000} \definecolor{Black}{rgb}{0.0,0.0,0.0} \definecolor{linkColor}{rgb}{0.0,0.0,0.554} \definecolor{citeColor}{rgb}{0.0,0.0,0.554} \definecolor{fileColor}{rgb}{0.0,0.0,0.554} \definecolor{urlColor}{rgb}{0.0,0.0,0.554} \definecolor{promptColor}{rgb}{0.0,0.0,0.589} \definecolor{brkpromptColor}{rgb}{0.589,0.0,0.0} \definecolor{gapinputColor}{rgb}{0.589,0.0,0.0} \definecolor{gapoutputColor}{rgb}{0.0,0.0,0.0} %% for a long time these were red and blue by default, %% now black, but keep variables to overwrite \definecolor{FuncColor}{rgb}{0.0,0.0,0.0} %% strange name because of pdflatex bug: \definecolor{Chapter }{rgb}{0.0,0.0,0.0} \definecolor{DarkOlive}{rgb}{0.1047,0.2412,0.0064} \usepackage{fancyvrb} \usepackage{mathptmx,helvet} \usepackage[T1]{fontenc} \usepackage{textcomp} \usepackage[ pdftex=true, bookmarks=true, a4paper=true, pdftitle={Written with GAPDoc}, pdfcreator={LaTeX with hyperref package / GAPDoc}, colorlinks=true, backref=page, breaklinks=true, linkcolor=linkColor, citecolor=citeColor, filecolor=fileColor, urlcolor=urlColor, pdfpagemode={UseNone}, ]{hyperref} \newcommand{\maintitlesize}{\fontsize{50}{55}\selectfont} % write page numbers to a .pnr log file for online help \newwrite\pagenrlog \immediate\openout\pagenrlog =\jobname.pnr \immediate\write\pagenrlog{PAGENRS := [} \newcommand{\logpage}[1]{\protect\write\pagenrlog{#1, \thepage,}} %% were never documented, give conflicts with some additional packages \newcommand{\GAP}{\textsf{GAP}} %% nicer description environments, allows long labels \usepackage{enumitem} \setdescription{style=nextline} %% depth of toc \setcounter{tocdepth}{1} %% command for ColorPrompt style examples \newcommand{\gapprompt}[1]{\color{promptColor}{\bfseries #1}} \newcommand{\gapbrkprompt}[1]{\color{brkpromptColor}{\bfseries #1}} \newcommand{\gapinput}[1]{\color{gapinputColor}{#1}} \begin{document} \logpage{[ 0, 0, 0 ]} \begin{titlepage} \mbox{}\vfill \begin{center}{\maintitlesize \textbf{ AutPGrp \\ \mbox{}}}\\ \vfill \hypersetup{pdftitle={ AutPGrp }} \markright{\scriptsize \mbox{}\hfill AutPGrp \hfill\mbox{}} {\Huge \textbf{ Computing the Automorphism Group of a p\texttt{\symbol{45}}Group \\ \mbox{}}}\\ \vfill {\Huge 1.12.0 \mbox{}}\\[1cm] { 17 May 2026 \mbox{}}\\[1cm] \mbox{}\\[2cm] {\Large \textbf{\strut Bettina Eick \strut\mbox{}}}\\ {\Large \textbf{\strut Eamonn O'Brien \strut\mbox{}}}\\ \hypersetup{pdfauthor={ Bettina Eick ; Eamonn O'Brien }} \end{center}\vfill \mbox{}\\ {\mbox{}\\ \small \noindent \textbf{ Bettina Eick } Email: \href{mailto://beick@tu-bs.de} {\texttt{beick@tu\texttt{\symbol{45}}bs.de}}\\ Homepage: \href{http://www.iaa.tu-bs.de/beick} {\texttt{http://www.iaa.tu\texttt{\symbol{45}}bs.de/beick}}\\ Address: \begin{minipage}[t]{8cm}\noindent Institut Analysis und Algebra\\ TU Braunschweig\\ Universit{\"a}tsplatz 2\\ D\texttt{\symbol{45}}38106 Braunschweig\\ Germany\\ \end{minipage} }\\ {\mbox{}\\ \small \noindent \textbf{ Eamonn O'Brien } Email: \href{mailto://obrien@math.auckland.ac.nz} {\texttt{obrien@math.auckland.ac.nz}}\\ Homepage: \href{http://www.math.auckland.ac.nz/~obrien} {\texttt{http://www.math.auckland.ac.nz/\texttt{\symbol{126}}obrien}}\\ Address: \begin{minipage}[t]{8cm}\noindent Department of Mathematics\\ University of Auckland\\ Private Bag 92019\\ Auckland\\ New Zealand\\ \end{minipage} }\\ \end{titlepage} \newpage\setcounter{page}{2} {\small \section*{Abstract} \logpage{[ 0, 0, 1 ]} The \textsf{AutPGrp} package introduces a new function to compute the automorphism group of a finite $p$\texttt{\symbol{45}}group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in \textsf{GAP} code and it makes use of a number of methods from the \textsf{GAP} library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in \textsf{GAP}. Our package usually out\texttt{\symbol{45}}performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. \mbox{}}\\[1cm] {\small \section*{Copyright} \logpage{[ 0, 0, 2 ]} Bettina Eick and Eamonn O'Brien. AutPGrp is free software; you can redistribute it and/or modify it under the terms of the \href{ https://www.fsf.org/licenses/gpl.html} {GNU General Public License} as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. \mbox{}}\\[1cm] {\small \section*{Acknowledgements} \logpage{[ 0, 0, 3 ]} We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the \textsf{GAP} \texttt{PQuotient} which are used in this package. \mbox{}}\\[1cm] \newpage \def\contentsname{Contents\logpage{[ 0, 0, 4 ]}} \tableofcontents \newpage \chapter{\textcolor{Chapter }{Introduction}}\label{Introduction} \logpage{[ 1, 0, 0 ]} \hyperdef{L}{X7DFB63A97E67C0A1}{} { \section{\textcolor{Chapter }{Introduction}}\logpage{[ 1, 1, 0 ]} \hyperdef{L}{X7DFB63A97E67C0A1}{} { Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the \textsf{GAP} 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a \emph{hybrid group} approach. More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the \textsf{GAP} 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of $p$\texttt{\symbol{45}}groups; the \textsf{GAP} 3 share package \textsf{Sisyphos} includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the \textsf{AutAg} share package of \textsf{GAP} 3. Moreover, the $p$\texttt{\symbol{45}}group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite $p$\texttt{\symbol{45}}group as outlined in O'Brien (1995). This algorithm is implemented in the ANU \texttt{pq} C program. Here we introduce a new function to compute the automorphism group of a finite $p$\texttt{\symbol{45}}group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU \texttt{pq} method. Our package is written in \textsf{GAP} code and it makes use of a number of methods from the \textsf{GAP} library such as the MeatAxe for matrix groups and permutation group functions. The \textsf{GAP} 4 package \textsf{ANUPQ}, which is an interface to most of the functionality of the ANU \texttt{pq} C program, uses the \textsf{AutPGrp} package to compute automorphism groups of $p$\texttt{\symbol{45}}groups. We have compared our method to the others available in \textsf{GAP}. Our package usually out\texttt{\symbol{45}}performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. } } \chapter{\textcolor{Chapter }{The automorphism group method}}\label{The automorphism group method} \logpage{[ 2, 0, 0 ]} \hyperdef{L}{X858BB98D85D78C8C}{} { \section{\textcolor{Chapter }{The automorphism group method}}\logpage{[ 2, 1, 0 ]} \hyperdef{L}{X858BB98D85D78C8C}{} { The \textsf{AutPGrp} package installs a method for \texttt{AutomorphismGroup} for a finite $p$\texttt{\symbol{45}}group (see also Section{\nobreakspace} (\textbf{Reference: Groups of Automorphisms}) in the \textsf{GAP} Reference Manual). \subsection{\textcolor{Chapter }{AutomorphismGroup}} \logpage{[ 2, 1, 1 ]}\nobreak \hyperdef{L}{X87677B0787B4461A}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AutomorphismGroup({\mdseries\slshape G})\index{AutomorphismGroup@\texttt{AutomorphismGroup}} \label{AutomorphismGroup} }\hfill{\scriptsize (operation)}}\\ \textbf{\indent Returns:\ } The automorphism group of \mbox{\texttt{\mdseries\slshape G}}. The input is a finite $p$\texttt{\symbol{45}}group \mbox{\texttt{\mdseries\slshape G}}. If the filters \texttt{IsPGroup}, \texttt{IsFinite} and \texttt{CanEasilyComputePcgs} are set and true for \mbox{\texttt{\mdseries\slshape G}}, the method selection of \textsf{GAP}{\nobreakspace}4 invokes this algorithm. The output of the method is an automorphism group, whose generators are given in \texttt{GroupHomomorphismByImages} format in terms of their action on the underlying group \mbox{\texttt{\mdseries\slshape G}}. } \subsection{\textcolor{Chapter }{InfoAutGrp}} \logpage{[ 2, 1, 2 ]}\nobreak \hyperdef{L}{X7D16AFE27E0B7133}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{InfoAutGrp\index{InfoAutGrp@\texttt{InfoAutGrp}} \label{InfoAutGrp} }\hfill{\scriptsize (info class)}}\\ This is a \textsf{GAP} \texttt{InfoClass} (\textbf{Reference: InfoClass for a GAP package}). By assigning an \texttt{level} in the range 1 to 4 via \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] SetInfoLevel(InfoAutGrp, level); \end{Verbatim} varying levels of information on the progress of the computation, will be obtained. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@LoadPackage("autpgrp", false);| true !gapprompt@gap>| !gapinput@G := PcGroupCode(619031068735, 32); # SmallGroup( 32, 15 );| !gapprompt@gap>| !gapinput@SetInfoLevel( InfoAutGrp, 1 );| !gapprompt@gap>| !gapinput@AutomorphismGroup(G);| #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 32 #I final step: convert \end{Verbatim} The algorithm proceeds by induction down the lower $p$\texttt{\symbol{45}}central series of \texttt{G} and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@G := DihedralGroup( IsPermGroup, 2^5 );;| !gapprompt@gap>| !gapinput@IsPGroup(G);| true !gapprompt@gap>| !gapinput@CanEasilyComputePcgs(G);| true !gapprompt@gap>| !gapinput@IsFinite(G);| true !gapprompt@gap>| !gapinput@A := AutomorphismGroup(G);| #I step 1: 2^2 -- init automorphisms #I step 2: 2^1 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 8 #I step 4: 2^1 -- aut grp has size 32 #I final step: convert !gapprompt@gap>| !gapinput@A.1;| Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) -> [ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ] !gapprompt@gap>| !gapinput@Order(A.1);| 16 \end{Verbatim} } } } \chapter{\textcolor{Chapter }{The underlying function}}\label{The underlying function} \logpage{[ 3, 0, 0 ]} \hyperdef{L}{X79DCF7D9853381F3}{} { \section{\textcolor{Chapter }{The underlying function}}\logpage{[ 3, 1, 0 ]} \hyperdef{L}{X79DCF7D9853381F3}{} { Underlying the method installation for \texttt{AutomorphismGroup} is the function \texttt{AutomorphismGroupPGroup}. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that \texttt{AutomorphismGroup} will always choose default values. \subsection{\textcolor{Chapter }{AutomorphismGroupPGroup}} \logpage{[ 3, 1, 1 ]}\nobreak \hyperdef{L}{X83ABE66C83446A0B}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{AutomorphismGroupPGroup({\mdseries\slshape G[, flag]})\index{AutomorphismGroupPGroup@\texttt{AutomorphismGroupPGroup}} \label{AutomorphismGroupPGroup} }\hfill{\scriptsize (function)}}\\ The input is a finite $p$\texttt{\symbol{45}}group as above and an optional \mbox{\texttt{\mdseries\slshape flag}} which can be true or false. Here the filters for \mbox{\texttt{\mdseries\slshape G}} need not be set, but they should be true for \mbox{\texttt{\mdseries\slshape G}}. The possible values for \mbox{\texttt{\mdseries\slshape flag}} are considered later in Chapter \ref{Influencing the algorithm}. If \mbox{\texttt{\mdseries\slshape flag}} is not supplied, the algorithm proceeds similarly to the method installed for \texttt{AutomorphismGroup}, but it produces slightly more detailed output. The output of the function is a record which contains the following fields: \begin{description} \item[{ \texttt{glAutos} }] a set of automorphisms which together with \texttt{agAutos} generate the automorphism group; \item[{ \texttt{glOrder} }] an integer whose product with the \texttt{agOrders} gives the size of the automorphism group; \item[{ \texttt{agAutos} }] a polycyclic generating sequence for a soluble normal subgroup of the automorphism group; \item[{ \texttt{agOrder} }] the relative orders corresponding to \texttt{agAutos}; \item[{ \texttt{one} }] the identity element of the automorphism group; \item[{ \texttt{group} }] the underlying group \mbox{\texttt{\mdseries\slshape G}}; \item[{ \texttt{size} }] the size of the automorphism group. \end{description} We do not return an automorphism group in the standard form because we wish to distinguish between \texttt{agAutos} and \texttt{glAutos}; the latter act non\texttt{\symbol{45}}trivially on the Frattini quotient of \mbox{\texttt{\mdseries\slshape G}}. This hybrid\texttt{\symbol{45}}group description of the automorphism group permits more efficient computations with it. The following function converts the output of \texttt{AutomorphismGroupPGroup} to the output of \texttt{AutomorphismGroup}. } \subsection{\textcolor{Chapter }{ConvertHybridAutGroup}} \logpage{[ 3, 1, 2 ]}\nobreak \hyperdef{L}{X7DEE06CF79DC7A7C}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ConvertHybridAutGroup({\mdseries\slshape A})\index{ConvertHybridAutGroup@\texttt{ConvertHybridAutGroup}} \label{ConvertHybridAutGroup} }\hfill{\scriptsize (function)}}\\ \textbf{\indent Returns:\ } A record. Let \mbox{\texttt{\mdseries\slshape A}} be the automorphism group of a $p$\texttt{\symbol{45}}group $G$ as computed by \texttt{AutomorphismGroupPGroup}. Then \texttt{ConvertHybridAutGroup} can compute a pc group isomorphic to the solvable part of \mbox{\texttt{\mdseries\slshape A}} stored in the record component \texttt{\mbox{\texttt{\mdseries\slshape A}}.agGroup}. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of $G$. The pc group facilitates various further computations with \mbox{\texttt{\mdseries\slshape A}}. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@LoadPackage("autpgrp", false);| true !gapprompt@gap>| !gapinput@H := PcGroupCode(297368117289422176, 729); # SmallGroup (729, 34);| !gapprompt@gap>| !gapinput@A := AutomorphismGroupPGroup(H);| #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert rec( agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ], agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [ ], glOper := [ ], glOrder := 1, group := , one := IdentityMapping( ), size := 52488 ) !gapprompt@gap>| !gapinput@ConvertHybridAutGroup( A );| \end{Verbatim} } \subsection{\textcolor{Chapter }{PcGroupAutPGroup}} \logpage{[ 3, 1, 3 ]}\nobreak \hyperdef{L}{X7ED1F7C6849CA658}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PcGroupAutPGroup({\mdseries\slshape A})\index{PcGroupAutPGroup@\texttt{PcGroupAutPGroup}} \label{PcGroupAutPGroup} }\hfill{\scriptsize (function)}}\\ This function computes a pc presentation for the solvable part of the automorphism group \mbox{\texttt{\mdseries\slshape A}} defined by \texttt{\mbox{\texttt{\mdseries\slshape A}}.agGroup}. \mbox{\texttt{\mdseries\slshape A}} is the output of the function \texttt{AutomorphismGroupPGroup}. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@H := PcGroupCode(297368117289422176, 729);; # SmallGroup (729, 34);| !gapprompt@gap>| !gapinput@A := AutomorphismGroupPGroup(H);;| #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert !gapprompt@gap>| !gapinput@B := PcGroupAutPGroup( A );| !gapprompt@gap>| !gapinput@I := InnerAutGroupPGroup( B );| Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, of ... ]) \end{Verbatim} } } } \chapter{\textcolor{Chapter }{Influencing the algorithm}}\label{Influencing the algorithm} \logpage{[ 4, 0, 0 ]} \hyperdef{L}{X86240F237D909E1C}{} { A number of choices can be made by the user to influence the performance of \texttt{AutomorphismGroupPGroup}. Below we identify these choices and their default values used in \texttt{AutomorphismGroup} (\ref{AutomorphismGroup}). We use the optional argument \mbox{\texttt{\mdseries\slshape flag}} of \texttt{AutomorphismGroupPGroup} to invoke user\texttt{\symbol{45}}defined choices. The possible values for \mbox{\texttt{\mdseries\slshape flag}} are \begin{description} \item[{\texttt{\mbox{\texttt{\mdseries\slshape flag}} = false} }] the user\texttt{\symbol{45}}defined defaults are employed in the algorithm. See below for a list of possibilities. \item[{\texttt{\mbox{\texttt{\mdseries\slshape flag}} = true} }] invokes the interactive version of the algorithm as described in Section \ref{An interactive version of the algorithm}. \end{description} In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter. \section{\textcolor{Chapter }{Outline of the algorithm}}\label{Outline of the algorithm} \logpage{[ 4, 1, 0 ]} \hyperdef{L}{X85D2ECC779E3CF7D}{} { The basic algorithm proceeds by induction down the lower $p$\texttt{\symbol{45}}central series of a given $p$\texttt{\symbol{45}}group $G$; that is, it successively computes $Aut(G_i)$ for the quotients $G_i = G / P_i(G)$ of the descending sequence of subgroups defined by $P_1(G) = G$ and $P_{i+1}(G)=[P_i(G),G] P_i(G)^p$ for $i\geq 1$. Hence, in the initial step of the algorithm, $Aut(G_2) = GL(d,p)$ where $d$ is the rank of the elementary abelian group $G_2$. In the inductive step it determines $Aut(G_{i+1})$ from $Aut(G_i)$. For this purpose we introduce an action of $Aut(G_i)$ on a certain elementary abelian $p$\texttt{\symbol{45}}group $M$ (the \emph{$p$\texttt{\symbol{45}}multiplicator} of $G_i$). The main computation of the inductive step is the determination of the stabiliser in $Aut(G_i)$ of a subgroup $U$ of $M$ (the \emph{allowable subgroup} for $G_{i+1}$). This stabiliser calculation is the bottle\texttt{\symbol{45}}neck of the algorithm. Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them. Observe that the initial step of the algorithm returns $GL(d,p)$. But $Aut(G)$ may induce on $G_2$ a proper subgroup, say $K$, of $GL(d,p)$. Any intermediate subgroup of $GL(d,p)$ which contains $K$ suffices for the algorithm and we supply two methods to construct a suitable subgroup: these use characteristic subgroups or invariants of normal subgroups of $G$. (See Section \ref{The initialisation step}.) In the inductive step an action of $Aut(G_i)$ on an elementary abelian group $M$ is used. This action is computed as a matrix action on a vector space. To simplify the orbit\texttt{\symbol{45}}stabiliser computation of the subspace $U$ of $M$, we can construct the stabiliser of $U$ by iteration over a sequence of $Aut(G_i)$\texttt{\symbol{45}}invariant subspaces of $M$. (See Section \ref{Stabilisers in matrix groups}.) Orbit\texttt{\symbol{45}}stabiliser computations in finite solvable groups given by a polycyclic generating sequence are much more efficient than generic computations of this type. Thus our algorithm makes use of a large solvable normal subgroup $S$ of $Aut(G_i)$. Further, it is useful if the generating set of $Aut(G_i)$ outside $S$ is as small as possible. To achieve this we determine a permutation representation of $Aut(G_i)/S$ and use this to reduce the number of generators if possible. (See Section \ref{Searching for a small generating set}.) } \section{\textcolor{Chapter }{The initialisation step}}\label{The initialisation step} \logpage{[ 4, 2, 0 ]} \hyperdef{L}{X87B27C1F810822F3}{} { Assume we seek to compute the automorphism group of a $p$\texttt{\symbol{45}}group $G$ having Frattini rank $d$. We first determine as small as possible a subgroup of $GL(d, p)$ whose extension can act on $G$. The user can choose the initialisation routine by assigning \texttt{InitAutGroup} to any one of the following: \begin{description} \item[{\texttt{InitAutomorphismGroupOver} }] to use the minimal overgroups: We determine the minimal over\texttt{\symbol{45}}groups of the Frattini subgroup of $G$ and compute invariants of these which must be respected by the automorphism group of $G$. We partition the minimal overgroups and compute the stabiliser in $GL(d, p)$ of this partition. The partition of the minimal overgroups is computed using the function \texttt{PGFingerprint( G, U )}. This is the time\texttt{\symbol{45}}consuming part of this initialisation method. The user can overwrite the function \texttt{PGFingerprint}. \item[{\texttt{InitAutomorphismGroupChar} }] to use the characteristic subgroups: Compute a generating set for the stabiliser in $GL (d, p)$ of a chain of characteristic subgroups of $G$. In practice, we construct a characteristic chain by determining 2\texttt{\symbol{45}}step centralisers and omega subgroups of factors of the lower $p$\texttt{\symbol{45}}central series. However, there are often other characteristic subgroups which are not found by these approaches. The user can overwrite the function \texttt{PGCharSubgroups( G )} to supply a set of characteristic subgroups. \item[{\texttt{InitAutomorphismGroupFull} }] to use the full $GL(d,p)$. \end{description} In the method for \texttt{AutomorphismGroup} (\ref{AutomorphismGroup}) we use a default strategy: if the value $\frac{p^d-1}{p-1}$ is less than 1000, then we use the minimal overgroup approach, otherwise the characteristic subgroups are employed. An exception is made for homogeneous abelian groups where we initialise the algorithm with the full group $GL(d,p)$. } \section{\textcolor{Chapter }{Stabilisers in matrix groups}}\label{Stabilisers in matrix groups} \logpage{[ 4, 3, 0 ]} \hyperdef{L}{X869B678180AD5E21}{} { Consider the $i$th inductive step of the algorithm. Here $A \leq Aut(G_i)$ acts as matrix group on the elementary abelian $p$\texttt{\symbol{45}}group $M$ and we want to determine the stabiliser of a subgroup $U \leq M$. We use the MeatAxe to compute a series of $A$\texttt{\symbol{45}}invariant subspaces through $M$ such that each factor in the series is irreducible as $A$\texttt{\symbol{45}}module. Then we use this series to break the computation of $Stab_A(U)$ into several smaller orbit\texttt{\symbol{45}}stabiliser calculations. Note that a theoretic argument yields an $A$\texttt{\symbol{45}}invariant subspace of $M$ a priori: the nucleus $N$. This is always used to split the computation up. However, it may happen that $N = M$ and hence results in no improvement. \subsection{\textcolor{Chapter }{CHOP{\textunderscore}MULT}} \logpage{[ 4, 3, 1 ]}\nobreak \hyperdef{L}{X86529C2080159D8A}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CHOP{\textunderscore}MULT\index{CHOP{\textunderscore}MULT@\texttt{CHOP{\textunderscore}MULT}} \label{CHOPuScoreMULT} }\hfill{\scriptsize (global variable)}}\\ The invariant series through $M$ is computed and used if the global variable \texttt{CHOP{\textunderscore}MULT} is set to \texttt{true}. Otherwise, the algorithm tries to determine $Stab_A(U)$ in one step. By default, \texttt{CHOP{\textunderscore}MULT} is \texttt{true}. } } \section{\textcolor{Chapter }{Searching for a small generating set}}\label{Searching for a small generating set} \logpage{[ 4, 4, 0 ]} \hyperdef{L}{X859CF9757A1F51EB}{} { After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor. If the automorphism group is soluble, we store a polycyclic generating set; otherwise, we store such a generating set for a large soluble normal subgroup $S$ of the automorphism group $A$, and as few generators outside as possible. If $S = A$ and a polycyclic generating set for $S$ is known, many steps of the algorithm proceed more rapidly. \subsection{\textcolor{Chapter }{NICE{\textunderscore}STAB}} \logpage{[ 4, 4, 1 ]}\nobreak \hyperdef{L}{X83C095D17D35A8DE}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NICE{\textunderscore}STAB\index{NICE{\textunderscore}STAB@\texttt{NICE{\textunderscore}STAB}} \label{NICEuScoreSTAB} }\hfill{\scriptsize (global variable)}}\\ It may be both time\texttt{\symbol{45}}consuming and difficult to reduce the number of generators for $A$ outside $S$. Note that if the initialisation of the algorithm is by \texttt{InitAutomorphismGroupOver}, then we always know a permutation representation for $A/S$. Occasionally the search for a small generating set is expensive. If this is observed, one could set the flag \texttt{NICE{\textunderscore}STAB} to \texttt{false} and the algorithm no longer invokes this search. } } \section{\textcolor{Chapter }{An interactive version of the algorithm}}\label{An interactive version of the algorithm} \logpage{[ 4, 5, 0 ]} \hyperdef{L}{X81A0235D7E15DFBD}{} { The choice of initialisation and the choice of chopping of the $p$\texttt{\symbol{45}}multiplicator can also be driven by an interactive version of the algorithm. We give an example. \begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single,label=Example] !gapprompt@gap>| !gapinput@G := PcGroupCode(2504972218445939562621471375, 256);; # SmallGroup( 2^8, 1000 );| !gapprompt@gap>| !gapinput@SetInfoLevel( InfoAutGrp, 3 );| !gapprompt@gap>| !gapinput@AutomorphismGroupPGroup( G, true );| #I step 1: 2^3 -- init automorphisms choose initialisation (Over/Char/Full): # we choose Full #I init automorphism group : Full #I step 2: 2^3 -- aut grp has size 168 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 3 dim N = 6 dim M = 6 chop M/N and N: (y/n): # we choose n #I induce autos and add central autos #I step 3: 2^2 -- aut grp has size 12288 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 6 dim N = 5 dim M = 8 chop M/N and N: (y/n): # we choose y #I induce autos and add central autos #I final step: convert rec( glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3, f4, f5, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4, agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ], agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], one := IdentityMapping( ), group := , size := 32768 ) \end{Verbatim} Two points are worthy of comment. First, the interactive version of the algorithm permits the user to make a suitable choice in each step of the algorithm instead of making one choice at the beginning. Secondly, the output of the \texttt{Info} function shows the ranks of the $p$\texttt{\symbol{45}}multiplicator and allowable subgroup, and thus allow the user to observe the scale of difficulty of the computation. } } \chapter{\textcolor{Chapter }{Additional features of the package}}\label{Additional features of the package} \logpage{[ 5, 0, 0 ]} \hyperdef{L}{X7C0BF58A85D48781}{} { \section{\textcolor{Chapter }{Additional features of the package}}\logpage{[ 5, 1, 0 ]} \hyperdef{L}{X7C0BF58A85D48781}{} { As an additional feature of this package we provide some functions to count extensions of $p$\texttt{\symbol{45}}groups and Lie algebras over $GF(p)$. These functions have been used in counting the $2$\texttt{\symbol{45}}groups of size $2^{10}$. \subsection{\textcolor{Chapter }{NumberOfPClass2PGroups (for 3 arguments)}} \logpage{[ 5, 1, 1 ]}\nobreak \hyperdef{L}{X7E472D2579705246}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfPClass2PGroups({\mdseries\slshape n, p, k})\index{NumberOfPClass2PGroups@\texttt{NumberOfPClass2PGroups}!for 3 arguments} \label{NumberOfPClass2PGroups:for 3 arguments} }\hfill{\scriptsize (operation)}}\\ This function determines the number of \mbox{\texttt{\mdseries\slshape n}}\texttt{\symbol{45}}generator \mbox{\texttt{\mdseries\slshape p}}\texttt{\symbol{45}}groups of \mbox{\texttt{\mdseries\slshape p}}\texttt{\symbol{45}}class 2 with Frattini subgroup of order \texttt{2\texttt{\symbol{94}}\mbox{\texttt{\mdseries\slshape k}}}. } \subsection{\textcolor{Chapter }{NumberOfPClass2PGroups (for 2 arguments)}} \logpage{[ 5, 1, 2 ]}\nobreak \hyperdef{L}{X825FC0D9796B0D97}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfPClass2PGroups({\mdseries\slshape n, p})\index{NumberOfPClass2PGroups@\texttt{NumberOfPClass2PGroups}!for 2 arguments} \label{NumberOfPClass2PGroups:for 2 arguments} }\hfill{\scriptsize (operation)}}\\ This function returns a list of of numbers of \mbox{\texttt{\mdseries\slshape n}}\texttt{\symbol{45}}generator \mbox{\texttt{\mdseries\slshape p}}\texttt{\symbol{45}}groups of \mbox{\texttt{\mdseries\slshape p}}\texttt{\symbol{45}}class 2 with Frattini subgroup of order \texttt{2\texttt{\symbol{94}}k} for \texttt{k} in $1, \ldots, n(n+1)/2$. } \subsection{\textcolor{Chapter }{NumberOfClass2LieAlgebras (for 3 arguments)}} \logpage{[ 5, 1, 3 ]}\nobreak \hyperdef{L}{X8399B9137982EBAC}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfClass2LieAlgebras({\mdseries\slshape n, p, k})\index{NumberOfClass2LieAlgebras@\texttt{NumberOfClass2LieAlgebras}!for 3 arguments} \label{NumberOfClass2LieAlgebras:for 3 arguments} }\hfill{\scriptsize (operation)}}\\ This function determines the number of \mbox{\texttt{\mdseries\slshape n}}\texttt{\symbol{45}}generator Lie algebras of class 2 over \mbox{\texttt{\mdseries\slshape GF(p)}} with derived Lie subalgebra of dimension \mbox{\texttt{\mdseries\slshape k}}. } \subsection{\textcolor{Chapter }{NumberOfClass2LieAlgbras (for 2 arguments)}} \logpage{[ 5, 1, 4 ]}\nobreak \hyperdef{L}{X824DFC307CA9626E}{} {\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfClass2LieAlgbras({\mdseries\slshape n, p})\index{NumberOfClass2LieAlgbras@\texttt{NumberOfClass2LieAlgbras}!for 2 arguments} \label{NumberOfClass2LieAlgbras:for 2 arguments} }\hfill{\scriptsize (operation)}}\\ This function returns a list of of numbers of \mbox{\texttt{\mdseries\slshape n}}\texttt{\symbol{45}}generator Lie algebras of class 2 over $GF(p)$ with derived Lie subalgebra of dimension $k$ for $k$ in $1, \ldots, n(n-1)/2$. } } } {\nobreakspace} \def\indexname{Index\logpage{[ "Ind", 0, 0 ]} \hyperdef{L}{X83A0356F839C696F}{} } \cleardoublepage \phantomsection \addcontentsline{toc}{chapter}{Index} \printindex \immediate\write\pagenrlog{["Ind", 0, 0], \arabic{page},} \newpage \immediate\write\pagenrlog{["End"], \arabic{page}];} \immediate\closeout\pagenrlog \end{document} autpgrp-1.12.0/doc/_main.xml0000644000000000000000000000073315202202400012543 0ustar00 ] > <#Include SYSTEM "title.xml"> <#Include SYSTEM "intro.xml"> <#Include SYSTEM "method.xml"> <#Include SYSTEM "underl.xml"> <#Include SYSTEM "influen.xml"> <#Include SYSTEM "additional.xml"> <#Include SYSTEM "_AutoDocMainFile.xml"> autpgrp-1.12.0/doc/additional.xml0000644000000000000000000000341415202202400013567 0ustar00 Additional features of the package

Additional features of the package As an additional feature of this package we provide some functions to count extensions of p-groups and Lie algebras over GF(p). These functions have been used in counting the 2-groups of size 2^{10}.

This function determines the number of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k. This function returns a list of of numbers of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k for k in 1, \ldots, n(n+1)/2. This function determines the number of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k. This function returns a list of of numbers of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k for k in 1, \ldots, n(n-1)/2.

autpgrp-1.12.0/doc/autpgrp.xml0000644000000000000000000000073415202202400013143 0ustar00 ] > <#Include SYSTEM "title.xml"> <#Include SYSTEM "intro.xml"> <#Include SYSTEM "method.xml"> <#Include SYSTEM "underl.xml"> <#Include SYSTEM "influen.xml"> <#Include SYSTEM "additional.xml"> <#Include SYSTEM "_AutoDocMainFile.xml"> autpgrp-1.12.0/doc/chap0.html0000644000000000000000000002142515202202400012620 0ustar00 GAP (AutPGrp) - Contents
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AutPGrp

Computing the Automorphism Group of a p-Group

1.12.0

17 May 2026

Bettina Eick
Email: beick@tu-bs.de
Homepage: http://www.iaa.tu-bs.de/beick
Address:
Institut Analysis und Algebra
TU Braunschweig
Universitätsplatz 2
D-38106 Braunschweig
Germany

Eamonn O'Brien
Email: obrien@math.auckland.ac.nz
Homepage: http://www.math.auckland.ac.nz/~obrien
Address:
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand

Abstract

The AutPGrp package introduces a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

Copyright

Bettina Eick and Eamonn O'Brien.

AutPGrp is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

Acknowledgements

We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the GAP PQuotient which are used in this package.

Contents

1 Introduction
 1.1 Introduction
2 The automorphism group method
 2.1 The automorphism group method

  2.1-1 AutomorphismGroup
  2.1-2 InfoAutGrp
3 The underlying function
 3.1 The underlying function

  3.1-1 AutomorphismGroupPGroup
  3.1-2 ConvertHybridAutGroup
  3.1-3 PcGroupAutPGroup
4 Influencing the algorithm
 4.1 Outline of the algorithm
 4.2 The initialisation step
 4.3 Stabilisers in matrix groups

  4.3-1 CHOP_MULT
 4.4 Searching for a small generating set

  4.4-1 NICE_STAB
 4.5 An interactive version of the algorithm
5 Additional features of the package
 5.1 Additional features of the package

  5.1-1 NumberOfPClass2PGroups
  5.1-2 NumberOfPClass2PGroups
  5.1-3 NumberOfClass2LieAlgebras
  5.1-4 NumberOfClass2LieAlgbras
Index

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autpgrp-1.12.0/doc/chap0.txt0000644000000000000000000001004615202202400012470 0ustar00  AutPGrp   Computing the Automorphism Group of a p-Group  1.12.0 17 May 2026 Bettina Eick Eamonn O'Brien Bettina Eick Email: mailto:beick@tu-bs.de Homepage: http://www.iaa.tu-bs.de/beick Address: Institut Analysis und Algebra TU Braunschweig Universitätsplatz 2 D-38106 Braunschweig Germany Eamonn O'Brien Email: mailto:obrien@math.auckland.ac.nz Homepage: http://www.math.auckland.ac.nz/~obrien Address: Department of Mathematics University of Auckland Private Bag 92019 Auckland New Zealand ------------------------------------------------------- Abstract The AutPGrp package introduces a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. ------------------------------------------------------- Copyright Bettina Eick and Eamonn O'Brien. AutPGrp is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License (https://www.fsf.org/licenses/gpl.html) as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. ------------------------------------------------------- Acknowledgements We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the GAP PQuotient which are used in this package. ------------------------------------------------------- Contents (AutPGrp) 1 Introduction 1.1 Introduction 2 The automorphism group method 2.1 The automorphism group method 2.1-1 AutomorphismGroup 2.1-2 InfoAutGrp 3 The underlying function 3.1 The underlying function 3.1-1 AutomorphismGroupPGroup 3.1-2 ConvertHybridAutGroup 3.1-3 PcGroupAutPGroup 4 Influencing the algorithm 4.1 Outline of the algorithm 4.2 The initialisation step 4.3 Stabilisers in matrix groups 4.3-1 CHOP_MULT 4.4 Searching for a small generating set 4.4-1 NICE_STAB 4.5 An interactive version of the algorithm 5 Additional features of the package 5.1 Additional features of the package 5.1-1 NumberOfPClass2PGroups 5.1-2 NumberOfPClass2PGroups 5.1-3 NumberOfClass2LieAlgebras 5.1-4 NumberOfClass2LieAlgbras  autpgrp-1.12.0/doc/chap0_mj.html0000644000000000000000000002204215202202400013302 0ustar00 GAP (AutPGrp) - Contents
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AutPGrp

Computing the Automorphism Group of a p-Group

1.12.0

17 May 2026

Bettina Eick
Email: beick@tu-bs.de
Homepage: http://www.iaa.tu-bs.de/beick
Address:
Institut Analysis und Algebra
TU Braunschweig
Universitätsplatz 2
D-38106 Braunschweig
Germany

Eamonn O'Brien
Email: obrien@math.auckland.ac.nz
Homepage: http://www.math.auckland.ac.nz/~obrien
Address:
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand

Abstract

The AutPGrp package introduces a new function to compute the automorphism group of a finite \(p\)-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

Copyright

Bettina Eick and Eamonn O'Brien.

AutPGrp is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

Acknowledgements

We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the GAP PQuotient which are used in this package.

Contents

1 Introduction
 1.1 Introduction
2 The automorphism group method
 2.1 The automorphism group method

  2.1-1 AutomorphismGroup
  2.1-2 InfoAutGrp
3 The underlying function
 3.1 The underlying function

  3.1-1 AutomorphismGroupPGroup
  3.1-2 ConvertHybridAutGroup
  3.1-3 PcGroupAutPGroup
4 Influencing the algorithm
 4.1 Outline of the algorithm
 4.2 The initialisation step
 4.3 Stabilisers in matrix groups

  4.3-1 CHOP_MULT
 4.4 Searching for a small generating set

  4.4-1 NICE_STAB
 4.5 An interactive version of the algorithm
5 Additional features of the package
 5.1 Additional features of the package

  5.1-1 NumberOfPClass2PGroups
  5.1-2 NumberOfPClass2PGroups
  5.1-3 NumberOfClass2LieAlgebras
  5.1-4 NumberOfClass2LieAlgbras
Index

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1 Introduction
 1.1 Introduction

1 Introduction

1.1 Introduction

Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the GAP 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a hybrid group approach.

More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the GAP 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of p-groups; the GAP 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of GAP 3.

Moreover, the p-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite p-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU pq C program.

Here we introduce a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU pq method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions.

The GAP 4 package ANUPQ, which is an interface to most of the functionality of the ANU pq C program, uses the AutPGrp package to compute automorphism groups of p-groups.

We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

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autpgrp-1.12.0/doc/chap1.txt0000644000000000000000000000555715202202400012504 0ustar00 1 Introduction 1.1 Introduction Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the GAP 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a hybrid group approach. More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the GAP 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of p-groups; the GAP 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of GAP 3. Moreover, the p-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite p-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU pq C program. Here we introduce a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU pq method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions. The GAP 4 package ANUPQ, which is an interface to most of the functionality of the ANU pq C program, uses the AutPGrp package to compute automorphism groups of p-groups. We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. autpgrp-1.12.0/doc/chap1_mj.html0000644000000000000000000001300015202202400013275 0ustar00 GAP (AutPGrp) - Chapter 1: Introduction
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1 Introduction
 1.1 Introduction

1 Introduction

1.1 Introduction

Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the GAP 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a hybrid group approach.

More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the GAP 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of \(p\)-groups; the GAP 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of GAP 3.

Moreover, the \(p\)-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite \(p\)-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU pq C program.

Here we introduce a new function to compute the automorphism group of a finite \(p\)-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU pq method. Our package is written in GAP code and it makes use of a number of methods from the GAP library such as the MeatAxe for matrix groups and permutation group functions.

The GAP 4 package ANUPQ, which is an interface to most of the functionality of the ANU pq C program, uses the AutPGrp package to compute automorphism groups of \(p\)-groups.

We have compared our method to the others available in GAP. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

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autpgrp-1.12.0/doc/chap2.html0000644000000000000000000001641515202202400012625 0ustar00 GAP (AutPGrp) - Chapter 2: The automorphism group method
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2 The automorphism group method
 2.1 The automorphism group method

  2.1-1 AutomorphismGroup
  2.1-2 InfoAutGrp

2 The automorphism group method

2.1 The automorphism group method

The AutPGrp package installs a method for AutomorphismGroup for a finite p-group (see also Section Reference: Groups of Automorphisms in the GAP Reference Manual).

2.1-1 AutomorphismGroup
‣ AutomorphismGroup( G )( operation )

Returns: The automorphism group of G.

The input is a finite p-group G. If the filters IsPGroup, IsFinite and CanEasilyComputePcgs are set and true for G, the method selection of GAP 4 invokes this algorithm.

The output of the method is an automorphism group, whose generators are given in GroupHomomorphismByImages format in terms of their action on the underlying group G.

2.1-2 InfoAutGrp
‣ InfoAutGrp( info class )

This is a GAP InfoClass (Reference: InfoClass for a GAP package). By assigning an level in the range 1 to 4 via

SetInfoLevel(InfoAutGrp, level);

varying levels of information on the progress of the computation, will be obtained.

gap> LoadPackage("autpgrp", false);
true
gap> G := PcGroupCode(619031068735, 32);  # SmallGroup( 32, 15 );
<pc group of size 32 with 5 generators>
gap> SetInfoLevel( InfoAutGrp, 1 );
gap> AutomorphismGroup(G);
#I  step 1: 2^2 -- init automorphisms
#I  step 2: 2^2 -- aut grp has size 2
#I  step 3: 2^1 -- aut grp has size 32
#I  final step: convert
<group of size 64 with 6 generators>

The algorithm proceeds by induction down the lower p-central series of G and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters.

gap> G := DihedralGroup( IsPermGroup, 2^5 );;
gap> IsPGroup(G);
true
gap> CanEasilyComputePcgs(G);
true
gap> IsFinite(G);
true
gap> A := AutomorphismGroup(G);
#I  step 1: 2^2 -- init automorphisms
#I  step 2: 2^1 -- aut grp has size 2
#I  step 3: 2^1 -- aut grp has size 8
#I  step 4: 2^1 -- aut grp has size 32
#I  final step: convert
<group of size 128 with 7 generators>
gap> A.1;
Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10),
  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) ->
[ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10),
  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]
gap> Order(A.1);
16
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autpgrp-1.12.0/doc/chap2.txt0000644000000000000000000001056215202202400012475 0ustar00 2 The automorphism group method 2.1 The automorphism group method The AutPGrp package installs a method for AutomorphismGroup for a finite p-group (see also Section 'Reference: Groups of Automorphisms' in the GAP Reference Manual). 2.1-1 AutomorphismGroup AutomorphismGroup( G )  operation Returns: The automorphism group of G. The input is a finite p-group G. If the filters IsPGroup, IsFinite and CanEasilyComputePcgs are set and true for G, the method selection of GAP 4 invokes this algorithm. The output of the method is an automorphism group, whose generators are given in GroupHomomorphismByImages format in terms of their action on the underlying group G. 2.1-2 InfoAutGrp InfoAutGrp  info class This is a GAP InfoClass (Reference: InfoClass for a GAP package). By assigning an level in the range 1 to 4 via  Example  SetInfoLevel(InfoAutGrp, level);  varying levels of information on the progress of the computation, will be obtained.  Example  gap> LoadPackage("autpgrp", false); true gap> G := PcGroupCode(619031068735, 32); # SmallGroup( 32, 15 );  gap> SetInfoLevel( InfoAutGrp, 1 ); gap> AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 32 #I final step: convert   The algorithm proceeds by induction down the lower p-central series of G and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters.  Example  gap> G := DihedralGroup( IsPermGroup, 2^5 );; gap> IsPGroup(G); true gap> CanEasilyComputePcgs(G); true gap> IsFinite(G); true gap> A := AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^1 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 8 #I step 4: 2^1 -- aut grp has size 32 #I final step: convert  gap> A.1; Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10),  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) -> [ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10),  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ] gap> Order(A.1); 16  autpgrp-1.12.0/doc/chap2_mj.html0000644000000000000000000001675415202202400013321 0ustar00 GAP (AutPGrp) - Chapter 2: The automorphism group method
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2 The automorphism group method
 2.1 The automorphism group method

  2.1-1 AutomorphismGroup
  2.1-2 InfoAutGrp

2 The automorphism group method

2.1 The automorphism group method

The AutPGrp package installs a method for AutomorphismGroup for a finite \(p\)-group (see also Section Reference: Groups of Automorphisms in the GAP Reference Manual).

2.1-1 AutomorphismGroup
‣ AutomorphismGroup( G )( operation )

Returns: The automorphism group of G.

The input is a finite \(p\)-group G. If the filters IsPGroup, IsFinite and CanEasilyComputePcgs are set and true for G, the method selection of GAP 4 invokes this algorithm.

The output of the method is an automorphism group, whose generators are given in GroupHomomorphismByImages format in terms of their action on the underlying group G.

2.1-2 InfoAutGrp
‣ InfoAutGrp( info class )

This is a GAP InfoClass (Reference: InfoClass for a GAP package). By assigning an level in the range 1 to 4 via

SetInfoLevel(InfoAutGrp, level);

varying levels of information on the progress of the computation, will be obtained.

gap> LoadPackage("autpgrp", false);
true
gap> G := PcGroupCode(619031068735, 32);  # SmallGroup( 32, 15 );
<pc group of size 32 with 5 generators>
gap> SetInfoLevel( InfoAutGrp, 1 );
gap> AutomorphismGroup(G);
#I  step 1: 2^2 -- init automorphisms
#I  step 2: 2^2 -- aut grp has size 2
#I  step 3: 2^1 -- aut grp has size 32
#I  final step: convert
<group of size 64 with 6 generators>

The algorithm proceeds by induction down the lower \(p\)-central series of G and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters.

gap> G := DihedralGroup( IsPermGroup, 2^5 );;
gap> IsPGroup(G);
true
gap> CanEasilyComputePcgs(G);
true
gap> IsFinite(G);
true
gap> A := AutomorphismGroup(G);
#I  step 1: 2^2 -- init automorphisms
#I  step 2: 2^1 -- aut grp has size 2
#I  step 3: 2^1 -- aut grp has size 8
#I  step 4: 2^1 -- aut grp has size 32
#I  final step: convert
<group of size 128 with 7 generators>
gap> A.1;
Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10),
  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) ->
[ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10),
  ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16),
  ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16),
  ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16),
  ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]
gap> Order(A.1);
16
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autpgrp-1.12.0/doc/chap3.html0000644000000000000000000002450715202202400012627 0ustar00 GAP (AutPGrp) - Chapter 3: The underlying function
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3 The underlying function
 3.1 The underlying function

  3.1-1 AutomorphismGroupPGroup
  3.1-2 ConvertHybridAutGroup
  3.1-3 PcGroupAutPGroup

3 The underlying function

3.1 The underlying function

Underlying the method installation for AutomorphismGroup is the function AutomorphismGroupPGroup. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that AutomorphismGroup will always choose default values.

3.1-1 AutomorphismGroupPGroup
‣ AutomorphismGroupPGroup( G[, flag] )( function )

The input is a finite p-group as above and an optional flag which can be true or false. Here the filters for G need not be set, but they should be true for G. The possible values for flag are considered later in Chapter 4. If flag is not supplied, the algorithm proceeds similarly to the method installed for AutomorphismGroup, but it produces slightly more detailed output. The output of the function is a record which contains the following fields:

glAutos

a set of automorphisms which together with agAutos generate the automorphism group;

glOrder

an integer whose product with the agOrders gives the size of the automorphism group;

agAutos

a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;

agOrder

the relative orders corresponding to agAutos;

one

the identity element of the automorphism group;

group

the underlying group G;

size

the size of the automorphism group.

We do not return an automorphism group in the standard form because we wish to distinguish between agAutos and glAutos; the latter act non-trivially on the Frattini quotient of G. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of AutomorphismGroupPGroup to the output of AutomorphismGroup.

3.1-2 ConvertHybridAutGroup
‣ ConvertHybridAutGroup( A )( function )

Returns: A record.

Let A be the automorphism group of a p-group G as computed by AutomorphismGroupPGroup. Then ConvertHybridAutGroup can compute a pc group isomorphic to the solvable part of A stored in the record component A.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of G. The pc group facilitates various further computations with A.

gap> LoadPackage("autpgrp", false);
true
gap> H := PcGroupCode(297368117289422176, 729);  # SmallGroup (729, 34);
<pc group of size 729 with 6 generators>
gap> A := AutomorphismGroupPGroup(H);
#I  step 1: 3^2 -- init automorphisms
#I  step 2: 3^1 -- aut grp has size 8
#I  step 3: 3^2 -- aut grp has size 72
#I  step 4: 3^1 -- aut grp has size 5832
#I  final step: convert
rec(
  agAutos :=
    [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5,
          f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
        [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5,
          f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
        [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ],
  agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [  ],
  glOper := [  ], glOrder := 1, group := <pc group of size 729 with
    6 generators>, one := IdentityMapping( <pc group of size 729 with
    6 generators> ), size := 52488 )
gap> ConvertHybridAutGroup( A );
<group of size 52488 with 11 generators>

3.1-3 PcGroupAutPGroup
‣ PcGroupAutPGroup( A )( function )

This function computes a pc presentation for the solvable part of the automorphism group A defined by A.agGroup. A is the output of the function AutomorphismGroupPGroup.

gap> H := PcGroupCode(297368117289422176, 729);;  # SmallGroup (729, 34);
gap> A := AutomorphismGroupPGroup(H);;
#I  step 1: 3^2 -- init automorphisms
#I  step 2: 3^1 -- aut grp has size 8
#I  step 3: 3^2 -- aut grp has size 72
#I  step 4: 3^1 -- aut grp has size 5832
#I  final step: convert
gap> B := PcGroupAutPGroup( A );
<pc group of size 52488 with 11 generators>
gap> I := InnerAutGroupPGroup( B );
Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, <identity> of ... ])
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autpgrp-1.12.0/doc/chap3.txt0000644000000000000000000001572115202202400012500 0ustar00 3 The underlying function 3.1 The underlying function Underlying the method installation for AutomorphismGroup is the function AutomorphismGroupPGroup. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that AutomorphismGroup will always choose default values. 3.1-1 AutomorphismGroupPGroup AutomorphismGroupPGroup( G[, flag] )  function The input is a finite p-group as above and an optional flag which can be true or false. Here the filters for G need not be set, but they should be true for G. The possible values for flag are considered later in Chapter 4. If flag is not supplied, the algorithm proceeds similarly to the method installed for AutomorphismGroup, but it produces slightly more detailed output. The output of the function is a record which contains the following fields:  glAutos  a set of automorphisms which together with agAutos generate the automorphism group;  glOrder  an integer whose product with the agOrders gives the size of the automorphism group;  agAutos  a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;  agOrder  the relative orders corresponding to agAutos;  one  the identity element of the automorphism group;  group  the underlying group G;  size  the size of the automorphism group. We do not return an automorphism group in the standard form because we wish to distinguish between agAutos and glAutos; the latter act non-trivially on the Frattini quotient of G. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of AutomorphismGroupPGroup to the output of AutomorphismGroup. 3.1-2 ConvertHybridAutGroup ConvertHybridAutGroup( A )  function Returns: A record. Let A be the automorphism group of a p-group G as computed by AutomorphismGroupPGroup. Then ConvertHybridAutGroup can compute a pc group isomorphic to the solvable part of A stored in the record component A.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of G. The pc group facilitates various further computations with A.  Example  gap> LoadPackage("autpgrp", false); true gap> H := PcGroupCode(297368117289422176, 729); # SmallGroup (729, 34);  gap> A := AutomorphismGroupPGroup(H); #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert rec(  agAutos :=  [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5,  f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->  [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5,  f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->  [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ],  Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ],  agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [ ],  glOper := [ ], glOrder := 1, group := , one := IdentityMapping( ), size := 52488 ) gap> ConvertHybridAutGroup( A );   3.1-3 PcGroupAutPGroup PcGroupAutPGroup( A )  function This function computes a pc presentation for the solvable part of the automorphism group A defined by A.agGroup. A is the output of the function AutomorphismGroupPGroup.  Example  gap> H := PcGroupCode(297368117289422176, 729);; # SmallGroup (729, 34); gap> A := AutomorphismGroupPGroup(H);; #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert gap> B := PcGroupAutPGroup( A );  gap> I := InnerAutGroupPGroup( B ); Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, of ... ])  autpgrp-1.12.0/doc/chap3_mj.html0000644000000000000000000002505215202202400013311 0ustar00 GAP (AutPGrp) - Chapter 3: The underlying function
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3 The underlying function
 3.1 The underlying function

  3.1-1 AutomorphismGroupPGroup
  3.1-2 ConvertHybridAutGroup
  3.1-3 PcGroupAutPGroup

3 The underlying function

3.1 The underlying function

Underlying the method installation for AutomorphismGroup is the function AutomorphismGroupPGroup. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that AutomorphismGroup will always choose default values.

3.1-1 AutomorphismGroupPGroup
‣ AutomorphismGroupPGroup( G[, flag] )( function )

The input is a finite \(p\)-group as above and an optional flag which can be true or false. Here the filters for G need not be set, but they should be true for G. The possible values for flag are considered later in Chapter 4. If flag is not supplied, the algorithm proceeds similarly to the method installed for AutomorphismGroup, but it produces slightly more detailed output. The output of the function is a record which contains the following fields:

glAutos

a set of automorphisms which together with agAutos generate the automorphism group;

glOrder

an integer whose product with the agOrders gives the size of the automorphism group;

agAutos

a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;

agOrder

the relative orders corresponding to agAutos;

one

the identity element of the automorphism group;

group

the underlying group G;

size

the size of the automorphism group.

We do not return an automorphism group in the standard form because we wish to distinguish between agAutos and glAutos; the latter act non-trivially on the Frattini quotient of G. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of AutomorphismGroupPGroup to the output of AutomorphismGroup.

3.1-2 ConvertHybridAutGroup
‣ ConvertHybridAutGroup( A )( function )

Returns: A record.

Let A be the automorphism group of a \(p\)-group \(G\) as computed by AutomorphismGroupPGroup. Then ConvertHybridAutGroup can compute a pc group isomorphic to the solvable part of A stored in the record component A.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of \(G\). The pc group facilitates various further computations with A.

gap> LoadPackage("autpgrp", false);
true
gap> H := PcGroupCode(297368117289422176, 729);  # SmallGroup (729, 34);
<pc group of size 729 with 6 generators>
gap> A := AutomorphismGroupPGroup(H);
#I  step 1: 3^2 -- init automorphisms
#I  step 2: 3^1 -- aut grp has size 8
#I  step 3: 3^2 -- aut grp has size 72
#I  step 4: 3^1 -- aut grp has size 5832
#I  final step: convert
rec(
  agAutos :=
    [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5,
          f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
        [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5,
          f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) ->
        [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ],
      Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ],
  agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [  ],
  glOper := [  ], glOrder := 1, group := <pc group of size 729 with
    6 generators>, one := IdentityMapping( <pc group of size 729 with
    6 generators> ), size := 52488 )
gap> ConvertHybridAutGroup( A );
<group of size 52488 with 11 generators>

3.1-3 PcGroupAutPGroup
‣ PcGroupAutPGroup( A )( function )

This function computes a pc presentation for the solvable part of the automorphism group A defined by A.agGroup. A is the output of the function AutomorphismGroupPGroup.

gap> H := PcGroupCode(297368117289422176, 729);;  # SmallGroup (729, 34);
gap> A := AutomorphismGroupPGroup(H);;
#I  step 1: 3^2 -- init automorphisms
#I  step 2: 3^1 -- aut grp has size 8
#I  step 3: 3^2 -- aut grp has size 72
#I  step 4: 3^1 -- aut grp has size 5832
#I  final step: convert
gap> B := PcGroupAutPGroup( A );
<pc group of size 52488 with 11 generators>
gap> I := InnerAutGroupPGroup( B );
Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, <identity> of ... ])
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autpgrp-1.12.0/doc/chap4.html0000644000000000000000000004453615202202400012634 0ustar00 GAP (AutPGrp) - Chapter 4: Influencing the algorithm
Goto Chapter: Top 1 2 3 4 5 Ind
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4 Influencing the algorithm
 4.1 Outline of the algorithm
 4.2 The initialisation step
 4.3 Stabilisers in matrix groups

  4.3-1 CHOP_MULT
 4.4 Searching for a small generating set

  4.4-1 NICE_STAB
 4.5 An interactive version of the algorithm

4 Influencing the algorithm

A number of choices can be made by the user to influence the performance of AutomorphismGroupPGroup. Below we identify these choices and their default values used in AutomorphismGroup (2.1-1). We use the optional argument flag of AutomorphismGroupPGroup to invoke user-defined choices. The possible values for flag are

flag = false

the user-defined defaults are employed in the algorithm. See below for a list of possibilities.

flag = true

invokes the interactive version of the algorithm as described in Section 4.5.

In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter.

4.1 Outline of the algorithm

The basic algorithm proceeds by induction down the lower p-central series of a given p-group G; that is, it successively computes Aut(G_i) for the quotients G_i = G / P_i(G) of the descending sequence of subgroups defined by P_1(G) = G and P_i+1(G)=[P_i(G),G] P_i(G)^p for i≥ 1. Hence, in the initial step of the algorithm, Aut(G_2) = GL(d,p) where d is the rank of the elementary abelian group G_2. In the inductive step it determines Aut(G_i+1) from Aut(G_i). For this purpose we introduce an action of Aut(G_i) on a certain elementary abelian p-group M (the p-multiplicator of G_i). The main computation of the inductive step is the determination of the stabiliser in Aut(G_i) of a subgroup U of M (the allowable subgroup for G_i+1). This stabiliser calculation is the bottle-neck of the algorithm.

Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them.

Observe that the initial step of the algorithm returns GL(d,p). But Aut(G) may induce on G_2 a proper subgroup, say K, of GL(d,p). Any intermediate subgroup of GL(d,p) which contains K suffices for the algorithm and we supply two methods to construct a suitable subgroup: these use characteristic subgroups or invariants of normal subgroups of G. (See Section 4.2.)

In the inductive step an action of Aut(G_i) on an elementary abelian group M is used. This action is computed as a matrix action on a vector space. To simplify the orbit-stabiliser computation of the subspace U of M, we can construct the stabiliser of U by iteration over a sequence of Aut(G_i)-invariant subspaces of M. (See Section 4.3.)

Orbit-stabiliser computations in finite solvable groups given by a polycyclic generating sequence are much more efficient than generic computations of this type. Thus our algorithm makes use of a large solvable normal subgroup S of Aut(G_i). Further, it is useful if the generating set of Aut(G_i) outside S is as small as possible. To achieve this we determine a permutation representation of Aut(G_i)/S and use this to reduce the number of generators if possible. (See Section 4.4.)

4.2 The initialisation step

Assume we seek to compute the automorphism group of a p-group G having Frattini rank d. We first determine as small as possible a subgroup of GL(d, p) whose extension can act on G.

The user can choose the initialisation routine by assigning InitAutGroup to any one of the following:

InitAutomorphismGroupOver

to use the minimal overgroups: We determine the minimal over-groups of the Frattini subgroup of G and compute invariants of these which must be respected by the automorphism group of G. We partition the minimal overgroups and compute the stabiliser in GL(d, p) of this partition.

The partition of the minimal overgroups is computed using the function PGFingerprint( G, U ). This is the time-consuming part of this initialisation method. The user can overwrite the function PGFingerprint.

InitAutomorphismGroupChar

to use the characteristic subgroups: Compute a generating set for the stabiliser in GL (d, p) of a chain of characteristic subgroups of G. In practice, we construct a characteristic chain by determining 2-step centralisers and omega subgroups of factors of the lower p-central series.

However, there are often other characteristic subgroups which are not found by these approaches. The user can overwrite the function PGCharSubgroups( G ) to supply a set of characteristic subgroups.

InitAutomorphismGroupFull

to use the full GL(d,p).

In the method for AutomorphismGroup (2.1-1) we use a default strategy: if the value fracp^d-1p-1 is less than 1000, then we use the minimal overgroup approach, otherwise the characteristic subgroups are employed. An exception is made for homogeneous abelian groups where we initialise the algorithm with the full group GL(d,p).

4.3 Stabilisers in matrix groups

Consider the ith inductive step of the algorithm. Here A ≤ Aut(G_i) acts as matrix group on the elementary abelian p-group M and we want to determine the stabiliser of a subgroup U ≤ M.

We use the MeatAxe to compute a series of A-invariant subspaces through M such that each factor in the series is irreducible as A-module. Then we use this series to break the computation of Stab_A(U) into several smaller orbit-stabiliser calculations.

Note that a theoretic argument yields an A-invariant subspace of M a priori: the nucleus N. This is always used to split the computation up. However, it may happen that N = M and hence results in no improvement.

4.3-1 CHOP_MULT
‣ CHOP_MULT( global variable )

The invariant series through M is computed and used if the global variable CHOP_MULT is set to true. Otherwise, the algorithm tries to determine Stab_A(U) in one step. By default, CHOP_MULT is true.

4.4 Searching for a small generating set

After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor.

If the automorphism group is soluble, we store a polycyclic generating set; otherwise, we store such a generating set for a large soluble normal subgroup S of the automorphism group A, and as few generators outside as possible. If S = A and a polycyclic generating set for S is known, many steps of the algorithm proceed more rapidly.

4.4-1 NICE_STAB
‣ NICE_STAB( global variable )

It may be both time-consuming and difficult to reduce the number of generators for A outside S. Note that if the initialisation of the algorithm is by InitAutomorphismGroupOver, then we always know a permutation representation for A/S. Occasionally the search for a small generating set is expensive. If this is observed, one could set the flag NICE_STAB to false and the algorithm no longer invokes this search.

4.5 An interactive version of the algorithm

The choice of initialisation and the choice of chopping of the p-multiplicator can also be driven by an interactive version of the algorithm. We give an example.

gap> G := PcGroupCode(2504972218445939562621471375, 256);;  # SmallGroup( 2^8, 1000 );
gap> SetInfoLevel( InfoAutGrp, 3 );

gap> AutomorphismGroupPGroup( G, true );
#I  step 1: 2^3 -- init automorphisms

choose initialisation (Over/Char/Full):     # we choose Full
#I    init automorphism group : Full
#I  step 2: 2^3 -- aut grp has size 168
#I    computing cover
#I    computing matrix action
#I    computing stabilizer of U
#I    dim U = 3  dim N = 6  dim M = 6

chop M/N and N: (y/n):                      # we choose n
#I    induce autos and add central autos
#I  step 3: 2^2 -- aut grp has size 12288
#I    computing cover
#I    computing matrix action
#I    computing stabilizer of U
#I    dim U = 6  dim N = 5  dim M = 8

chop M/N and N: (y/n):                      # we choose y
#I    induce autos and add central autos
#I  final step: convert
rec(
  glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3,
          f4, f5, f6*f7, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4,
  agAutos :=
    [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5,
          f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ],
  agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
  one := IdentityMapping( <pc group of size 256 with 8 generators> ),
  group := <pc group of size 256 with 8 generators>, size := 32768 )

Two points are worthy of comment. First, the interactive version of the algorithm permits the user to make a suitable choice in each step of the algorithm instead of making one choice at the beginning. Secondly, the output of the Info function shows the ranks of the p-multiplicator and allowable subgroup, and thus allow the user to observe the scale of difficulty of the computation.

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autpgrp-1.12.0/doc/chap4.txt0000644000000000000000000003377315202202400012510 0ustar00 4 Influencing the algorithm A number of choices can be made by the user to influence the performance of AutomorphismGroupPGroup. Below we identify these choices and their default values used in AutomorphismGroup (2.1-1). We use the optional argument flag of AutomorphismGroupPGroup to invoke user-defined choices. The possible values for flag are flag = false  the user-defined defaults are employed in the algorithm. See below for a list of possibilities. flag = true  invokes the interactive version of the algorithm as described in Section 4.5. In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter. 4.1 Outline of the algorithm The basic algorithm proceeds by induction down the lower p-central series of a given p-group G; that is, it successively computes Aut(G_i) for the quotients G_i = G / P_i(G) of the descending sequence of subgroups defined by P_1(G) = G and P_i+1(G)=[P_i(G),G] P_i(G)^p for i≥ 1. Hence, in the initial step of the algorithm, Aut(G_2) = GL(d,p) where d is the rank of the elementary abelian group G_2. In the inductive step it determines Aut(G_i+1) from Aut(G_i). For this purpose we introduce an action of Aut(G_i) on a certain elementary abelian p-group M (the p-multiplicator of G_i). The main computation of the inductive step is the determination of the stabiliser in Aut(G_i) of a subgroup U of M (the allowable subgroup for G_i+1). This stabiliser calculation is the bottle-neck of the algorithm. Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them. Observe that the initial step of the algorithm returns GL(d,p). But Aut(G) may induce on G_2 a proper subgroup, say K, of GL(d,p). Any intermediate subgroup of GL(d,p) which contains K suffices for the algorithm and we supply two methods to construct a suitable subgroup: these use characteristic subgroups or invariants of normal subgroups of G. (See Section 4.2.) In the inductive step an action of Aut(G_i) on an elementary abelian group M is used. This action is computed as a matrix action on a vector space. To simplify the orbit-stabiliser computation of the subspace U of M, we can construct the stabiliser of U by iteration over a sequence of Aut(G_i)-invariant subspaces of M. (See Section 4.3.) Orbit-stabiliser computations in finite solvable groups given by a polycyclic generating sequence are much more efficient than generic computations of this type. Thus our algorithm makes use of a large solvable normal subgroup S of Aut(G_i). Further, it is useful if the generating set of Aut(G_i) outside S is as small as possible. To achieve this we determine a permutation representation of Aut(G_i)/S and use this to reduce the number of generators if possible. (See Section 4.4.) 4.2 The initialisation step Assume we seek to compute the automorphism group of a p-group G having Frattini rank d. We first determine as small as possible a subgroup of GL(d, p) whose extension can act on G. The user can choose the initialisation routine by assigning InitAutGroup to any one of the following: InitAutomorphismGroupOver  to use the minimal overgroups: We determine the minimal over-groups of the Frattini subgroup of G and compute invariants of these which must be respected by the automorphism group of G. We partition the minimal overgroups and compute the stabiliser in GL(d, p) of this partition. The partition of the minimal overgroups is computed using the function PGFingerprint( G, U ). This is the time-consuming part of this initialisation method. The user can overwrite the function PGFingerprint. InitAutomorphismGroupChar  to use the characteristic subgroups: Compute a generating set for the stabiliser in GL (d, p) of a chain of characteristic subgroups of G. In practice, we construct a characteristic chain by determining 2-step centralisers and omega subgroups of factors of the lower p-central series. However, there are often other characteristic subgroups which are not found by these approaches. The user can overwrite the function PGCharSubgroups( G ) to supply a set of characteristic subgroups. InitAutomorphismGroupFull  to use the full GL(d,p). In the method for AutomorphismGroup (2.1-1) we use a default strategy: if the value fracp^d-1p-1 is less than 1000, then we use the minimal overgroup approach, otherwise the characteristic subgroups are employed. An exception is made for homogeneous abelian groups where we initialise the algorithm with the full group GL(d,p). 4.3 Stabilisers in matrix groups Consider the ith inductive step of the algorithm. Here A ≤ Aut(G_i) acts as matrix group on the elementary abelian p-group M and we want to determine the stabiliser of a subgroup U ≤ M. We use the MeatAxe to compute a series of A-invariant subspaces through M such that each factor in the series is irreducible as A-module. Then we use this series to break the computation of Stab_A(U) into several smaller orbit-stabiliser calculations. Note that a theoretic argument yields an A-invariant subspace of M a priori: the nucleus N. This is always used to split the computation up. However, it may happen that N = M and hence results in no improvement. 4.3-1 CHOP_MULT CHOP_MULT  global variable The invariant series through M is computed and used if the global variable CHOP_MULT is set to true. Otherwise, the algorithm tries to determine Stab_A(U) in one step. By default, CHOP_MULT is true. 4.4 Searching for a small generating set After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor. If the automorphism group is soluble, we store a polycyclic generating set; otherwise, we store such a generating set for a large soluble normal subgroup S of the automorphism group A, and as few generators outside as possible. If S = A and a polycyclic generating set for S is known, many steps of the algorithm proceed more rapidly. 4.4-1 NICE_STAB NICE_STAB  global variable It may be both time-consuming and difficult to reduce the number of generators for A outside S. Note that if the initialisation of the algorithm is by InitAutomorphismGroupOver, then we always know a permutation representation for A/S. Occasionally the search for a small generating set is expensive. If this is observed, one could set the flag NICE_STAB to false and the algorithm no longer invokes this search. 4.5 An interactive version of the algorithm The choice of initialisation and the choice of chopping of the p-multiplicator can also be driven by an interactive version of the algorithm. We give an example.  Example  gap> G := PcGroupCode(2504972218445939562621471375, 256);; # SmallGroup( 2^8, 1000 ); gap> SetInfoLevel( InfoAutGrp, 3 );  gap> AutomorphismGroupPGroup( G, true ); #I step 1: 2^3 -- init automorphisms  choose initialisation (Over/Char/Full): # we choose Full #I init automorphism group : Full #I step 2: 2^3 -- aut grp has size 168 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 3 dim N = 6 dim M = 6  chop M/N and N: (y/n): # we choose n #I induce autos and add central autos #I step 3: 2^2 -- aut grp has size 12288 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 6 dim N = 5 dim M = 8  chop M/N and N: (y/n): # we choose y #I induce autos and add central autos #I final step: convert rec(  glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3,  f4, f5, f6*f7, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4,  agAutos :=  [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5,  f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ],  Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->  [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ],  agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],  one := IdentityMapping( ),  group := , size := 32768 )  Two points are worthy of comment. First, the interactive version of the algorithm permits the user to make a suitable choice in each step of the algorithm instead of making one choice at the beginning. Secondly, the output of the Info function shows the ranks of the p-multiplicator and allowable subgroup, and thus allow the user to observe the scale of difficulty of the computation. autpgrp-1.12.0/doc/chap4_mj.html0000644000000000000000000004562715202202400013324 0ustar00 GAP (AutPGrp) - Chapter 4: Influencing the algorithm
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4 Influencing the algorithm
 4.1 Outline of the algorithm
 4.2 The initialisation step
 4.3 Stabilisers in matrix groups

  4.3-1 CHOP_MULT
 4.4 Searching for a small generating set

  4.4-1 NICE_STAB
 4.5 An interactive version of the algorithm

4 Influencing the algorithm

A number of choices can be made by the user to influence the performance of AutomorphismGroupPGroup. Below we identify these choices and their default values used in AutomorphismGroup (2.1-1). We use the optional argument flag of AutomorphismGroupPGroup to invoke user-defined choices. The possible values for flag are

flag = false

the user-defined defaults are employed in the algorithm. See below for a list of possibilities.

flag = true

invokes the interactive version of the algorithm as described in Section 4.5.

In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter.

4.1 Outline of the algorithm

The basic algorithm proceeds by induction down the lower \(p\)-central series of a given \(p\)-group \(G\); that is, it successively computes \(Aut(G_i)\) for the quotients \(G_i = G / P_i(G)\) of the descending sequence of subgroups defined by \(P_1(G) = G\) and \(P_{i+1}(G)=[P_i(G),G] P_i(G)^p\) for \(i\geq 1\). Hence, in the initial step of the algorithm, \(Aut(G_2) = GL(d,p)\) where \(d\) is the rank of the elementary abelian group \(G_2\). In the inductive step it determines \(Aut(G_{i+1})\) from \(Aut(G_i)\). For this purpose we introduce an action of \(Aut(G_i)\) on a certain elementary abelian \(p\)-group \(M\) (the \(p\)-multiplicator of \(G_i\)). The main computation of the inductive step is the determination of the stabiliser in \(Aut(G_i)\) of a subgroup \(U\) of \(M\) (the allowable subgroup for \(G_{i+1}\)). This stabiliser calculation is the bottle-neck of the algorithm.

Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them.

Observe that the initial step of the algorithm returns \(GL(d,p)\). But \(Aut(G)\) may induce on \(G_2\) a proper subgroup, say \(K\), of \(GL(d,p)\). Any intermediate subgroup of \(GL(d,p)\) which contains \(K\) suffices for the algorithm and we supply two methods to construct a suitable subgroup: these use characteristic subgroups or invariants of normal subgroups of \(G\). (See Section 4.2.)

In the inductive step an action of \(Aut(G_i)\) on an elementary abelian group \(M\) is used. This action is computed as a matrix action on a vector space. To simplify the orbit-stabiliser computation of the subspace \(U\) of \(M\), we can construct the stabiliser of \(U\) by iteration over a sequence of \(Aut(G_i)\)-invariant subspaces of \(M\). (See Section 4.3.)

Orbit-stabiliser computations in finite solvable groups given by a polycyclic generating sequence are much more efficient than generic computations of this type. Thus our algorithm makes use of a large solvable normal subgroup \(S\) of \(Aut(G_i)\). Further, it is useful if the generating set of \(Aut(G_i)\) outside \(S\) is as small as possible. To achieve this we determine a permutation representation of \(Aut(G_i)/S\) and use this to reduce the number of generators if possible. (See Section 4.4.)

4.2 The initialisation step

Assume we seek to compute the automorphism group of a \(p\)-group \(G\) having Frattini rank \(d\). We first determine as small as possible a subgroup of \(GL(d, p)\) whose extension can act on \(G\).

The user can choose the initialisation routine by assigning InitAutGroup to any one of the following:

InitAutomorphismGroupOver

to use the minimal overgroups: We determine the minimal over-groups of the Frattini subgroup of \(G\) and compute invariants of these which must be respected by the automorphism group of \(G\). We partition the minimal overgroups and compute the stabiliser in \(GL(d, p)\) of this partition.

The partition of the minimal overgroups is computed using the function PGFingerprint( G, U ). This is the time-consuming part of this initialisation method. The user can overwrite the function PGFingerprint.

InitAutomorphismGroupChar

to use the characteristic subgroups: Compute a generating set for the stabiliser in \(GL (d, p)\) of a chain of characteristic subgroups of \(G\). In practice, we construct a characteristic chain by determining 2-step centralisers and omega subgroups of factors of the lower \(p\)-central series.

However, there are often other characteristic subgroups which are not found by these approaches. The user can overwrite the function PGCharSubgroups( G ) to supply a set of characteristic subgroups.

InitAutomorphismGroupFull

to use the full \(GL(d,p)\).

In the method for AutomorphismGroup (2.1-1) we use a default strategy: if the value \(\frac{p^d-1}{p-1}\) is less than 1000, then we use the minimal overgroup approach, otherwise the characteristic subgroups are employed. An exception is made for homogeneous abelian groups where we initialise the algorithm with the full group \(GL(d,p)\).

4.3 Stabilisers in matrix groups

Consider the \(i\)th inductive step of the algorithm. Here \(A \leq Aut(G_i)\) acts as matrix group on the elementary abelian \(p\)-group \(M\) and we want to determine the stabiliser of a subgroup \(U \leq M\).

We use the MeatAxe to compute a series of \(A\)-invariant subspaces through \(M\) such that each factor in the series is irreducible as \(A\)-module. Then we use this series to break the computation of \(Stab_A(U)\) into several smaller orbit-stabiliser calculations.

Note that a theoretic argument yields an \(A\)-invariant subspace of \(M\) a priori: the nucleus \(N\). This is always used to split the computation up. However, it may happen that \(N = M\) and hence results in no improvement.

4.3-1 CHOP_MULT
‣ CHOP_MULT( global variable )

The invariant series through \(M\) is computed and used if the global variable CHOP_MULT is set to true. Otherwise, the algorithm tries to determine \(Stab_A(U)\) in one step. By default, CHOP_MULT is true.

4.4 Searching for a small generating set

After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor.

If the automorphism group is soluble, we store a polycyclic generating set; otherwise, we store such a generating set for a large soluble normal subgroup \(S\) of the automorphism group \(A\), and as few generators outside as possible. If \(S = A\) and a polycyclic generating set for \(S\) is known, many steps of the algorithm proceed more rapidly.

4.4-1 NICE_STAB
‣ NICE_STAB( global variable )

It may be both time-consuming and difficult to reduce the number of generators for \(A\) outside \(S\). Note that if the initialisation of the algorithm is by InitAutomorphismGroupOver, then we always know a permutation representation for \(A/S\). Occasionally the search for a small generating set is expensive. If this is observed, one could set the flag NICE_STAB to false and the algorithm no longer invokes this search.

4.5 An interactive version of the algorithm

The choice of initialisation and the choice of chopping of the \(p\)-multiplicator can also be driven by an interactive version of the algorithm. We give an example.

gap> G := PcGroupCode(2504972218445939562621471375, 256);;  # SmallGroup( 2^8, 1000 );
gap> SetInfoLevel( InfoAutGrp, 3 );

gap> AutomorphismGroupPGroup( G, true );
#I  step 1: 2^3 -- init automorphisms

choose initialisation (Over/Char/Full):     # we choose Full
#I    init automorphism group : Full
#I  step 2: 2^3 -- aut grp has size 168
#I    computing cover
#I    computing matrix action
#I    computing stabilizer of U
#I    dim U = 3  dim N = 6  dim M = 6

chop M/N and N: (y/n):                      # we choose n
#I    induce autos and add central autos
#I  step 3: 2^2 -- aut grp has size 12288
#I    computing cover
#I    computing matrix action
#I    computing stabilizer of U
#I    dim U = 6  dim N = 5  dim M = 8

chop M/N and N: (y/n):                      # we choose y
#I    induce autos and add central autos
#I  final step: convert
rec(
  glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3,
          f4, f5, f6*f7, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4,
  agAutos :=
    [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5,
          f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ],
      Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) ->
        [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ],
  agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
  one := IdentityMapping( <pc group of size 256 with 8 generators> ),
  group := <pc group of size 256 with 8 generators>, size := 32768 )

Two points are worthy of comment. First, the interactive version of the algorithm permits the user to make a suitable choice in each step of the algorithm instead of making one choice at the beginning. Secondly, the output of the Info function shows the ranks of the \(p\)-multiplicator and allowable subgroup, and thus allow the user to observe the scale of difficulty of the computation.

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5 Additional features of the package
 5.1 Additional features of the package

  5.1-1 NumberOfPClass2PGroups
  5.1-2 NumberOfPClass2PGroups
  5.1-3 NumberOfClass2LieAlgebras
  5.1-4 NumberOfClass2LieAlgbras

5 Additional features of the package

5.1 Additional features of the package

As an additional feature of this package we provide some functions to count extensions of p-groups and Lie algebras over GF(p). These functions have been used in counting the 2-groups of size 2^10.

5.1-1 NumberOfPClass2PGroups
‣ NumberOfPClass2PGroups( n, p, k )( operation )

This function determines the number of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k.

5.1-2 NumberOfPClass2PGroups
‣ NumberOfPClass2PGroups( n, p )( operation )

This function returns a list of of numbers of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k for k in 1, ..., n(n+1)/2.

5.1-3 NumberOfClass2LieAlgebras
‣ NumberOfClass2LieAlgebras( n, p, k )( operation )

This function determines the number of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k.

5.1-4 NumberOfClass2LieAlgbras
‣ NumberOfClass2LieAlgbras( n, p )( operation )

This function returns a list of of numbers of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k for k in 1, ..., n(n-1)/2.

 

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autpgrp-1.12.0/doc/chap5.txt0000644000000000000000000000362315202202400012500 0ustar00 5 Additional features of the package 5.1 Additional features of the package As an additional feature of this package we provide some functions to count extensions of p-groups and Lie algebras over GF(p). These functions have been used in counting the 2-groups of size 2^10. 5.1-1 NumberOfPClass2PGroups NumberOfPClass2PGroups( n, p, k )  operation This function determines the number of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k. 5.1-2 NumberOfPClass2PGroups NumberOfPClass2PGroups( n, p )  operation This function returns a list of of numbers of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k for k in 1, ..., n(n+1)/2. 5.1-3 NumberOfClass2LieAlgebras NumberOfClass2LieAlgebras( n, p, k )  operation This function determines the number of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k. 5.1-4 NumberOfClass2LieAlgbras NumberOfClass2LieAlgbras( n, p )  operation This function returns a list of of numbers of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k for k in 1, ..., n(n-1)/2.   autpgrp-1.12.0/doc/chap5_mj.html0000644000000000000000000001430515202202400013312 0ustar00 GAP (AutPGrp) - Chapter 5: Additional features of the package
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5 Additional features of the package
 5.1 Additional features of the package

  5.1-1 NumberOfPClass2PGroups
  5.1-2 NumberOfPClass2PGroups
  5.1-3 NumberOfClass2LieAlgebras
  5.1-4 NumberOfClass2LieAlgbras

5 Additional features of the package

5.1 Additional features of the package

As an additional feature of this package we provide some functions to count extensions of \(p\)-groups and Lie algebras over \(GF(p)\). These functions have been used in counting the \(2\)-groups of size \(2^{10}\).

5.1-1 NumberOfPClass2PGroups
‣ NumberOfPClass2PGroups( n, p, k )( operation )

This function determines the number of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k.

5.1-2 NumberOfPClass2PGroups
‣ NumberOfPClass2PGroups( n, p )( operation )

This function returns a list of of numbers of n-generator p-groups of p-class 2 with Frattini subgroup of order 2^k for k in \(1, \ldots, n(n+1)/2\).

5.1-3 NumberOfClass2LieAlgebras
‣ NumberOfClass2LieAlgebras( n, p, k )( operation )

This function determines the number of n-generator Lie algebras of class 2 over GF(p) with derived Lie subalgebra of dimension k.

5.1-4 NumberOfClass2LieAlgbras
‣ NumberOfClass2LieAlgbras( n, p )( operation )

This function returns a list of of numbers of n-generator Lie algebras of class 2 over \(GF(p)\) with derived Lie subalgebra of dimension \(k\) for \(k\) in \(1, \ldots, n(n-1)/2\).

 

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Index

AutomorphismGroup 2.1-1
AutomorphismGroupPGroup 3.1-1
CHOP_MULT 4.3-1
ConvertHybridAutGroup 3.1-2
InfoAutGrp 2.1-2
NICE_STAB 4.4-1
NumberOfClass2LieAlgbras, for 2 arguments 5.1-4
NumberOfClass2LieAlgebras, for 3 arguments 5.1-3
NumberOfPClass2PGroups, for 2 arguments 5.1-2
    for 3 arguments 5.1-1
PcGroupAutPGroup 3.1-3

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autpgrp-1.12.0/doc/chapInd.txt0000644000000000000000000000104315202202400013040 0ustar00 Index AutomorphismGroup 2.1-1 AutomorphismGroupPGroup 3.1-1 CHOP_MULT 4.3-1 ConvertHybridAutGroup 3.1-2 InfoAutGrp 2.1-2 NICE_STAB 4.4-1 NumberOfClass2LieAlgbras, for 2 arguments 5.1-4 NumberOfClass2LieAlgebras, for 3 arguments 5.1-3 NumberOfPClass2PGroups, for 2 arguments 5.1-2 for 3 arguments 5.1-1 PcGroupAutPGroup 3.1-3 ------------------------------------------------------- autpgrp-1.12.0/doc/chapInd_mj.html0000644000000000000000000000633615202202400013665 0ustar00 GAP (AutPGrp) - Index
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Index

AutomorphismGroup 2.1-1
AutomorphismGroupPGroup 3.1-1
CHOP_MULT 4.3-1
ConvertHybridAutGroup 3.1-2
InfoAutGrp 2.1-2
NICE_STAB 4.4-1
NumberOfClass2LieAlgbras, for 2 arguments 5.1-4
NumberOfClass2LieAlgebras, for 3 arguments 5.1-3
NumberOfPClass2PGroups, for 2 arguments 5.1-2
    for 3 arguments 5.1-1
PcGroupAutPGroup 3.1-3

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autpgrp-1.12.0/doc/dark.css0000644000000000000000000000515115202202400012370 0ustar00/* dark.css Frank Luebeck */ /* Initial dark theme contributed by kiryph in issue #75. */ /* colors */ body { background: #121212; color: #eee; } a:link { color: #576cad; } a:visited { color: #576cad; } a:active { color: #eee; } a:hover { background: #eee; } pre { color: black; } tt, code { color: #eee; } /* layout for the definitions of functions, variables, ... */ div.func { background: #909090; } /* Example elements (for old converted manuals, now in div+pre */ table.example { background: #efefef; } /* becomes ... */ div.example { background: #efefef; color: black; } /* Links to chapters in all files at top and bottom. */ div.chlinktop { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinktop a:hover { background: #2f3742; } div.chlinkbot { background: #22272e; border-color: #3a414a; color: #d7dde5; } /* and this is for the "Top", "Prev", "Next" links */ div.chlinkprevnexttop { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinkprevnexttop a:hover { background: #2f3742; } div.chlinkprevnextbot { background: #22272e; border-color: #3a414a; color: #d7dde5; } div.chlinkprevnextbot a:hover { background: #2f3742; } div.ContChap div.ContSect:hover div.ContSSBlock { background: #eee; border-color: #666; color: #000; } div.ContSSBlock a:hover { background: #fff; } /* and here for the side menu of contents in the chapter files */ div.ChapSects a:hover { background: #eee; color: #000; } div.ChapSects div.ContSect:hover div.ContSSBlock { background: #b5b5b5; border-color: #666; color: #000; } div.ChapSects div.ContSect:hover div.ContSSBlock a:hover { background: #828282; } /* Table elements */ table.GAPDocTable { border-color: black; } table.GAPDocTable td, table.GAPDocTable th { border-color: #555; } table.GAPDocTablenoborder td, table.GAPDocTable th { border-color: #555; } /* Colors and fonts can be overwritten for some types of elements. */ /* Verb elements */ pre.normal { color: #eee; } /* Func-like elements and Ref to Func-like */ code.func { color: #eee; } /* K elements */ code.keyw { color: #983d3d; } /* F elements */ code.file { color: #8e4510; } /* Arg elements */ var.Arg { color: #060; } /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000097; } span.GAPbrkprompt { color: #970000; } span.GAPinput { color: #970000; } /* Bib entries */ span.BibKey { color: #052; } /* for light and dark mode pictures */ .only-on-dark { display: block; } .only-on-light { display: none; } autpgrp-1.12.0/doc/influen.xml0000644000000000000000000002701315202202400013120 0ustar00 Influencing the algorithm A number of choices can be made by the user to influence the performance of AutomorphismGroupPGroup. Below we identify these choices and their default values used in . We use the optional argument flag of AutomorphismGroupPGroup to invoke user-defined choices. The possible values for flag are

flag = false the user-defined defaults are employed in the algorithm. See below for a list of possibilities. flag = true invokes the interactive version of the algorithm as described in Section .

In the next section we give a brief outline of the algorithm which is necessary to understand the possible choices. Then we introduce the choices the later sections of this chapter.

Outline of the algorithm The basic algorithm proceeds by induction down the lower p-central series of a given p-group G; that is, it successively computes Aut(G_i) for the quotients G_i = G / P_i(G) of the descending sequence of subgroups defined by P_1(G) = G and P_{i+1}(G)=[P_i(G),G] P_i(G)^p for i\geq 1. Hence, in the initial step of the algorithm, Aut(G_2) = GL(d,p) where d is the rank of the elementary abelian group G_2. In the inductive step it determines Aut(G_{i+1}) from Aut(G_i). For this purpose we introduce an action of Aut(G_i) on a certain elementary abelian p-group M (the p-multiplicator of G_i). The main computation of the inductive step is the determination of the stabiliser in Aut(G_i) of a subgroup U of M (the allowable subgroup for G_{i+1}). This stabiliser calculation is the bottle-neck of the algorithm.

Our package incorporates a number of refinements designed to simplify this stabiliser computation. Some of these refinements incur overheads and hence they are not always invoked. The features outlined below allow the user to select them.

Observe that the initial step of the algorithm returns GL(d,p). But Aut(G) may induce on G_2 a proper subgroup, say K, of GL(d,p). Any intermediate subgroup of GL(d,p) which contains K suffices for the algorithm and we supply two methods to construct a suitable subgroup: these use characteristic subgroups or invariants of normal subgroups of G. (See Section .)

In the inductive step an action of Aut(G_i) on an elementary abelian group M is used. This action is computed as a matrix action on a vector space. To simplify the orbit-stabiliser computation of the subspace U of M, we can construct the stabiliser of U by iteration over a sequence of Aut(G_i)-invariant subspaces of M. (See Section .)

Orbit-stabiliser computations in finite solvable groups given by a polycyclic generating sequence are much more efficient than generic computations of this type. Thus our algorithm makes use of a large solvable normal subgroup S of Aut(G_i). Further, it is useful if the generating set of Aut(G_i) outside S is as small as possible. To achieve this we determine a permutation representation of Aut(G_i)/S and use this to reduce the number of generators if possible. (See Section .)

The initialisation step Assume we seek to compute the automorphism group of a p-group G having Frattini rank d. We first determine as small as possible a subgroup of GL(d, p) whose extension can act on G.

The user can choose the initialisation routine by assigning InitAutGroup to any one of the following: InitAutomorphismGroupOver to use the minimal overgroups: We determine the minimal over-groups of the Frattini subgroup of G and compute invariants of these which must be respected by the automorphism group of G. We partition the minimal overgroups and compute the stabiliser in GL(d, p) of this partition.

The partition of the minimal overgroups is computed using the function PGFingerprint( G, U ). This is the time-consuming part of this initialisation method. The user can overwrite the function PGFingerprint. InitAutomorphismGroupChar to use the characteristic subgroups: Compute a generating set for the stabiliser in GL (d, p) of a chain of characteristic subgroups of G. In practice, we construct a characteristic chain by determining 2-step centralisers and omega subgroups of factors of the lower p-central series.

However, there are often other characteristic subgroups which are not found by these approaches. The user can overwrite the function PGCharSubgroups( G ) to supply a set of characteristic subgroups. InitAutomorphismGroupFull to use the full GL(d,p). In the method for we use a default strategy: if the value \frac{p^d-1}{p-1} is less than 1000, then we use the minimal overgroup approach, otherwise the characteristic subgroups are employed. An exception is made for homogeneous abelian groups where we initialise the algorithm with the full group GL(d,p).

Stabilisers in matrix groups Consider the ith inductive step of the algorithm. Here A \leq Aut(G_i) acts as matrix group on the elementary abelian p-group M and we want to determine the stabiliser of a subgroup U \leq M.

We use the MeatAxe to compute a series of A-invariant subspaces through M such that each factor in the series is irreducible as A-module. Then we use this series to break the computation of Stab_A(U) into several smaller orbit-stabiliser calculations.

Note that a theoretic argument yields an A-invariant subspace of M a priori: the nucleus N. This is always used to split the computation up. However, it may happen that N = M and hence results in no improvement. The invariant series through M is computed and used if the global variable CHOP_MULT is set to true. Otherwise, the algorithm tries to determine Stab_A(U) in one step. By default, CHOP_MULT is true.

Searching for a small generating set After each step of the computation, we attempt to find a nice generating set for the automorphism group of the current factor.

If the automorphism group is soluble, we store a polycyclic generating set; otherwise, we store such a generating set for a large soluble normal subgroup S of the automorphism group A, and as few generators outside as possible. If S = A and a polycyclic generating set for S is known, many steps of the algorithm proceed more rapidly. It may be both time-consuming and difficult to reduce the number of generators for A outside S. Note that if the initialisation of the algorithm is by InitAutomorphismGroupOver, then we always know a permutation representation for A/S. Occasionally the search for a small generating set is expensive. If this is observed, one could set the flag NICE_STAB to false and the algorithm no longer invokes this search.

An interactive version of the algorithm The choice of initialisation and the choice of chopping of the p-multiplicator can also be driven by an interactive version of the algorithm. We give an example.

G := PcGroupCode(2504972218445939562621471375, 256);; # SmallGroup( 2^8, 1000 ); gap> SetInfoLevel( InfoAutGrp, 3 ); gap> AutomorphismGroupPGroup( G, true ); #I step 1: 2^3 -- init automorphisms choose initialisation (Over/Char/Full): # we choose Full #I init automorphism group : Full #I step 2: 2^3 -- aut grp has size 168 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 3 dim N = 6 dim M = 6 chop M/N and N: (y/n): # we choose n #I induce autos and add central autos #I step 3: 2^2 -- aut grp has size 12288 #I computing cover #I computing matrix action #I computing stabilizer of U #I dim U = 6 dim N = 5 dim M = 8 chop M/N and N: (y/n): # we choose y #I induce autos and add central autos #I final step: convert rec( glAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f3, f3, f4, f5, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3*f5*f6, f2*f3, f3, f4, f5*f8, f6*f7, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f3, f2*f4, f3, f4*f7, f5*f7, f6*f7*f8, f7, f8 ] ], glOrder := 4, agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f4, f2, f3, f4*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f4, f3, f4*f7, f5, f6*f7*f8, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f5, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f5, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f5, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f6, f2, f3, f4, f5*f7*f8, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f6, f3, f4*f7*f8, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f8, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f8, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f8, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1*f7, f2, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2*f7, f3, f4, f5, f6, f7, f8 ], Pcgs([ f1, f2, f3, f4, f5, f6, f7, f8 ]) -> [ f1, f2, f3*f7, f4, f5, f6, f7, f8 ] ], agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], one := IdentityMapping( ), group := , size := 32768 ) ]]>

Two points are worthy of comment. First, the interactive version of the algorithm permits the user to make a suitable choice in each step of the algorithm instead of making one choice at the beginning. Secondly, the output of the Info function shows the ranks of the p-multiplicator and allowable subgroup, and thus allow the user to observe the scale of difficulty of the computation.

autpgrp-1.12.0/doc/intro.xml0000644000000000000000000000512715202202400012615 0ustar00 Introduction
Introduction Given an arbitrary finite group, the computation of its automorphism group is a very difficult task. Pioneer work in this area was carried out by Felsch and Neubueser (1970), whose algorithm used the output of their subgroup lattice program. A technique developed by Neubueser in the early 1970s sought to compute the automorphism group viewed as a permutation group acting on unions of certain conjugacy classes of the group. A similar method was implemented by Hulpke (1997) in the &GAP; 4 library. Recently, Cannon and Holt (1999) presented a new algorithm which uses a hybrid group approach.

More efficient approaches are available to determine the automorphism group for groups satisfying certain properties. Following the work of Shoda (1928), Hulpke in 1997 implemented a practical method for finite abelian groups in the &GAP; 4 library. Wursthorn (1993) adapted modular group algebra techniques to compute the automorphism groups of p-groups; the &GAP; 3 share package Sisyphos includes an implementation. Smith (1994) introduced an algorithm for finite solvable groups which is available in the AutAg share package of &GAP; 3.

Moreover, the p-group generation method of Newman (1977) and O'Brien (1990) can be modified to compute the automorphism group of a finite p-group as outlined in O'Brien (1995). This algorithm is implemented in the ANU pq C program.

Here we introduce a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANU pq method. Our package is written in &GAP; code and it makes use of a number of methods from the &GAP; library such as the MeatAxe for matrix groups and permutation group functions.

The &GAP; 4 package ANUPQ, which is an interface to most of the functionality of the ANU pq C program, uses the &AUTPGRP; package to compute automorphism groups of p-groups.

We have compared our method to the others available in &GAP;. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed.

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The automorphism group method The &AUTPGRP; package installs a method for AutomorphismGroup for a finite p-group (see also Section  in the &GAP; Reference Manual). The automorphism group of G. The input is a finite p-group G. If the filters IsPGroup, IsFinite and CanEasilyComputePcgs are set and true for G, the method selection of &GAP; 4 invokes this algorithm.

The output of the method is an automorphism group, whose generators are given in GroupHomomorphismByImages format in terms of their action on the underlying group G. This is a &GAP; . By assigning an level in the range 1 to 4 via SetInfoLevel(InfoAutGrp, level); varying levels of information on the progress of the computation, will be obtained.

LoadPackage("autpgrp", false); true gap> G := PcGroupCode(619031068735, 32); # SmallGroup( 32, 15 ); gap> SetInfoLevel( InfoAutGrp, 1 ); gap> AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 32 #I final step: convert ]]>

The algorithm proceeds by induction down the lower p-central series of G and the information corresponds to the steps of this induction. In the following example we observe that the method also accepts permutation groups as input, provided they satisfy the required filters.

G := DihedralGroup( IsPermGroup, 2^5 );; gap> IsPGroup(G); true gap> CanEasilyComputePcgs(G); true gap> IsFinite(G); true gap> A := AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^1 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 8 #I step 4: 2^1 -- aut grp has size 32 #I final step: convert gap> A.1; Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) -> [ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ] gap> Order(A.1); 16 ]]>

autpgrp-1.12.0/doc/nocolorprompt.css0000644000000000000000000000031315202202400014357 0ustar00 /* colors for ColorPrompt like examples */ span.GAPprompt { color: #000000; font-weight: normal; } span.GAPbrkprompt { color: #000000; font-weight: normal; } span.GAPinput { color: #000000; } autpgrp-1.12.0/doc/ragged.css0000644000000000000000000000023115202202400012672 0ustar00/* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { text-align: left; } autpgrp-1.12.0/doc/rainbow.js0000644000000000000000000000533615202202400012741 0ustar00 function randchar(str) { var i = Math.floor(Math.random() * str.length); while (i == str.length) i = Math.floor(Math.random() * str.length); return str[i]; } hexdigits = "0123456789abcdef"; function randlight() { return randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits)+ randchar("cdef")+randchar(hexdigits) } function randdark() { return randchar("012345789")+randchar(hexdigits)+ randchar("012345789")+randchar(hexdigits)+ randchar("102345789")+randchar(hexdigits) } document.write('\n'); autpgrp-1.12.0/doc/times.css0000644000000000000000000000026115202202400012565 0ustar00/* times.css Frank Lübeck */ /* Change default CSS to use Times font. */ body { font-family: Times,Times New Roman,serif; } autpgrp-1.12.0/doc/title.xml0000644000000000000000000000472415202202400012605 0ustar00 AutPGrp Computing the Automorphism Group of a p-Group 1.12.0 Bettina Eick
Institut Analysis und Algebra
TU Braunschweig
Universitätsplatz 2
D-38106 Braunschweig
Germany
beick@tu-bs.de http://www.iaa.tu-bs.de/beick Eamonn O'Brien
Department of Mathematics
University of Auckland
Private Bag 92019
Auckland
New Zealand
obrien@math.auckland.ac.nz http://www.math.auckland.ac.nz/~obrien 17 May 2026 The &AUTPGRP; package introduces a new function to compute the automorphism group of a finite p-group. The underlying algorithm is a refinement of the methods described in O'Brien (1995). In particular, this implementation is more efficient in both time and space requirements and hence has a wider range of applications than the ANUPQ method. Our package is written in &GAP; code and it makes use of a number of methods from the &GAP; library such as the MeatAxe for matrix groups and permutation group functions. We have compared our method to the others available in &GAP;. Our package usually out-performs all but the method designed for finite abelian groups. We note that our method uses the small groups library in certain cases and hence our algorithm is more effective if the small groups library is installed. Bettina Eick and Eamonn O'Brien.

AutPGrp is free software; you can redistribute it and/or modify it under the terms of the https://www.fsf.org/licenses/gpl.html as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. We thank Alexander Hulpke for helping us with efficiency problems. Werner Nickel provided some functions from the &GAP; PQuotient which are used in this package. autpgrp-1.12.0/doc/toggless.css0000644000000000000000000000167215202202400013302 0ustar00/* toggless.css Frank Lübeck */ /* Using javascript we change all div.ContSect to div.ContSectOpen or div.ContSectClosed. This way the config for div.ContSect in manual.css is no longer relevant. Here we add the CSS for the new elements. */ /* This layout is based on an idea by Burkhard Höfling. */ div.ContSectClosed { text-align: left; margin-left: 1em; } div.ContSectOpen { text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock { display: block; text-align: left; margin-left: 1em; } div.ContSectOpen div.ContSSBlock a { display: block; width: 100%; margin-left: 1em; } span.tocline a:hover { display: inline; background: #eeeeee; } span.ContSS a:hover { display: inline; background: #eeeeee; } span.toctoggle { font-size: 80%; display: inline-block; width: 1.2em; } span.toctoggle:hover { background-color: #aaaaaa; } autpgrp-1.12.0/doc/toggless.js0000644000000000000000000000420515202202400013121 0ustar00/* toggless.js Frank Lübeck */ /* this file contains two functions: mergeSideTOCHooks: this changes div.ContSect elements to the class ContSectClosed and includes a hook to toggle between ContSectClosed and ContSectOpen. openclosetoc: this function does the toggling, the rest is done by CSS */ closedTOCMarker = "▶ "; openTOCMarker = "▼ "; noTOCMarker = " "; /* merge hooks into side toc for opening/closing subsections with openclosetoc */ function mergeSideTOCHooks() { var hlist = document.getElementsByTagName("div"); for (var i = 0; i < hlist.length; i++) { if (hlist[i].className == "ContSect") { var chlds = hlist[i].childNodes; var el = document.createElement("span"); var oncl = document.createAttribute("class"); oncl.nodeValue = "toctoggle"; el.setAttributeNode(oncl); var cont; if (chlds.length > 2) { var oncl = document.createAttribute("onclick"); oncl.nodeValue = "openclosetoc(event)"; el.setAttributeNode(oncl); cont = document.createTextNode(closedTOCMarker); } else { cont = document.createTextNode(noTOCMarker); } el.appendChild(cont); hlist[i].firstChild.insertBefore(el, hlist[i].firstChild.firstChild); hlist[i].className = "ContSectClosed"; } } } function openclosetoc (event) { /* first two steps to make it work in most browsers */ var evt=window.event || event; if (!evt.target) evt.target=evt.srcElement; var markClosed = document.createTextNode(closedTOCMarker); var markOpen = document.createTextNode(openTOCMarker); var par = evt.target.parentNode.parentNode; if (par.className == "ContSectOpen") { par.className = "ContSectClosed"; evt.target.replaceChild(markClosed, evt.target.firstChild); } else if (par.className == "ContSectClosed") { par.className = "ContSectOpen"; evt.target.replaceChild(markOpen, evt.target.firstChild); } } /* adjust jscontent which is called onload */ jscontentfuncs.push(mergeSideTOCHooks); autpgrp-1.12.0/doc/underl.xml0000644000000000000000000001325715202202400012756 0ustar00 The underlying function

The underlying function Underlying the method installation for AutomorphismGroup is the function AutomorphismGroupPGroup. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that AutomorphismGroup will always choose default values. The input is a finite p-group as above and an optional flag which can be true or false. Here the filters for G need not be set, but they should be true for G. The possible values for flag are considered later in Chapter . If flag is not supplied, the algorithm proceeds similarly to the method installed for AutomorphismGroup, but it produces slightly more detailed output. The output of the function is a record which contains the following fields:

glAutos a set of automorphisms which together with agAutos generate the automorphism group; glOrder an integer whose product with the agOrders gives the size of the automorphism group; agAutos a polycyclic generating sequence for a soluble normal subgroup of the automorphism group; agOrder the relative orders corresponding to agAutos; one the identity element of the automorphism group; group the underlying group G; size the size of the automorphism group.

We do not return an automorphism group in the standard form because we wish to distinguish between agAutos and glAutos; the latter act non-trivially on the Frattini quotient of G. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of AutomorphismGroupPGroup to the output of AutomorphismGroup. A record. Let A be the automorphism group of a p-group G as computed by AutomorphismGroupPGroup. Then can compute a pc group isomorphic to the solvable part of A stored in the record component A.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of G. The pc group facilitates various further computations with A. LoadPackage("autpgrp", false); true gap> H := PcGroupCode(297368117289422176, 729); # SmallGroup (729, 34); gap> A := AutomorphismGroupPGroup(H); #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert rec( agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ], agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [ ], glOper := [ ], glOrder := 1, group := , one := IdentityMapping( ), size := 52488 ) gap> ConvertHybridAutGroup( A ); ]]> This function computes a pc presentation for the solvable part of the automorphism group A defined by A.agGroup. A is the output of the function AutomorphismGroupPGroup. H := PcGroupCode(297368117289422176, 729);; # SmallGroup (729, 34); gap> A := AutomorphismGroupPGroup(H);; #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert gap> B := PcGroupAutPGroup( A ); gap> I := InnerAutGroupPGroup( B ); Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, of ... ]) ]]>

autpgrp-1.12.0/gap/0000755000000000000000000000000015202202400010735 5ustar00autpgrp-1.12.0/gap/autoops.gi0000644000000000000000000001542315202202400012755 0ustar00############################################################################# ## #W autoops.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F PGAutomorphism( , , ) ## InstallMethod( PGAutomorphism, "for p-groups", true, [IsPGroup and IsFinite, IsList, IsList ], 0, function( G, gens, imgs ) local filter, type, r, p, pcgs, base, pcgsimgs, baseimgs, def, d; # cache the default type in the group if not IsBound( G!.PGAutomType ) then filter := IsPGAutomorphism and IsBijective; type := TypeOfDefaultGeneralMapping( G, G, filter ); G!.PGAutomType := type; else type := G!.PGAutomType; fi; # get images correct r := RankPGroup( G ); p := PrimePGroup( G ); pcgs := Pcgs( G ); base := pcgs{[1..r]}; if gens = pcgs then pcgsimgs := imgs; baseimgs := imgs{[1..r]}; elif gens = base then baseimgs := ShallowCopy( imgs ); pcgsimgs := ShallowCopy( imgs ); for d in G!.definitions do if not IsNegRat( d ) then Add( pcgsimgs, SubstituteDef( d, pcgsimgs, p ) ); fi; od; else Print("# W computing pcgs in PGAutomorphism \n"); pcgsimgs := CanonicalPcgsByGeneratorsWithImages( pcgs, gens, imgs ); baseimgs := pcgsimgs{[1..r]}; fi; # create homomorphism return Objectify( type, rec( pcgs := pcgs, pcgsimgs := pcgsimgs, base := base, baseimgs := baseimgs ) ); end); ############################################################################# ## #F IdentityPGAutomorphism( ) ## InstallGlobalFunction( IdentityPGAutomorphism, function( G ) return PGAutomorphism( G, Pcgs(G), AsList(Pcgs(G)) ); end ); ############################################################################# ## #F PrintObj(auto) ## InstallMethod( PrintObj, "for group automorphisms", true, [IsPGAutomorphism], SUM_FLAGS, function( auto ) if IsBound( auto!.mat ) then Print("Aut + Mat: ",auto!.pcgsimgs); else Print("Aut: ",auto!.pcgsimgs); fi; end); ############################################################################# ## #F ViewObj(auto) ## InstallMethod( ViewObj, "for group automorphisms", true, [IsPGAutomorphism], SUM_FLAGS, function( auto ) if IsBound( auto!.mat ) then Print("Aut + Mat: ",auto!.pcgsimgs); else Print("Aut: ",auto!.pcgsimgs); fi; end); ############################################################################# ## #F \= ## InstallMethod( \=, "for group automorphisms", IsIdenticalObj, [IsPGAutomorphism, IsPGAutomorphism], 0, function( auto1, auto2 ) return auto1!.base = auto2!.base and auto1!.baseimgs = auto2!.baseimgs; end); ############################################################################# ## #F ImagesRepresentative( auto, g ) ## InstallMethod( ImagesRepresentative, "for group automorphisms", true, [IsPGAutomorphism, IsObject], 0, function( auto, g ) return MappedPcElement( g, auto!.pcgs, auto!.pcgsimgs ); end ); ############################################################################# ## #F PGMult( auto1, auto2 ) ## InstallMethod( PGMult, true, [IsPGAutomorphism, IsPGAutomorphism], 0, function( auto1, auto2 ) local new, aut; # 1. version new := List( auto1!.pcgsimgs, x -> ImagesRepresentative( auto2, x ) ); if IsBound( auto1!.mat ) and IsBound( auto2!.mat ) then aut := PGAutomorphism( Source(auto1), auto1!.pcgs, new ); aut!.mat := auto1!.mat * auto2!.mat; else aut := PGAutomorphism( Source(auto1), auto1!.pcgs, new ); fi; return aut; # 2. version new := List( auto2!.baseimgs, x -> ImagesRepresentative( auto1, x ) ); return PGAutomorphism( Source(auto2), auto2!.base, new ); end ); ############################################################################# ## #F CompositionMapping2( auto1, auto2 ) ## InstallMethod( CompositionMapping2, "for group automorphisms", true, [IsPGAutomorphism, IsPGAutomorphism], 0, function( auto2, auto1 ) return PGMult( auto1, auto2 ); end ); ############################################################################# ## #F PGMultList( autl ) ## InstallMethod( PGMultList, true, [IsList], 0, function( autl ) local l, r, new; if Length( autl ) = 1 then return autl[1]; fi; if Length( autl ) = 2 then return PGMult(autl[1],autl[2]); fi; l := Length( autl ); r := QuoInt( l, 2 ); new := List( [1..r], x -> PGMult( autl[2*x-1], autl[2*x] )); if not IsInt( l/2 ) then Add( new, autl[l] ); fi; return PGMultList( new ); end ); ############################################################################# ## #F PGPower( n, aut ) ## InstallMethod( PGPower, true, [IsInt, IsPGAutomorphism], 0, function( n, aut ) local c, l, i, j, new; if n <= 0 then return fail; fi; if n = 1 then return aut; fi; c := CoefficientsQadic( n, 2 ); # create power list, if necessary if not IsBound( aut!.power ) then aut!.power := []; fi; # add powers, if necessary l := Length( aut!.power ); if l = 0 then new := aut; else new := aut!.power[l]; fi; for i in [l+1..Length(c)-1] do new := PGMult( new, new ); Add( aut!.power, new ); od; # multiply powers together if c[1] = 1 then new := [aut]; else new := []; fi; for i in [2..Length(c)] do if c[i] = 1 then Add( new, aut!.power[i-1] ); fi; od; return PGMultList( new ); end ); ############################################################################# ## #F PGInverse( aut ) ## InstallMethod( PGInverse, true, [IsPGAutomorphism], 0, function( aut ) local new, inv; if not IsBound( aut!.inv ) then new := CanonicalPcgsByGeneratorsWithImages( aut!.pcgs, aut!.pcgsimgs, aut!.pcgs); inv := PGAutomorphism( Source( aut ), aut!.pcgs, new[2] ); else inv := aut!.inv; fi; if IsBound( aut!.mat ) and not IsBound( inv!.mat) then inv!.mat := aut!.mat^-1; fi; aut!.inv := inv; return aut!.inv; end ); ############################################################################# ## #F InverseGeneralMapping(auto) ## InstallOtherMethod( InverseGeneralMapping, "for group automorphism", true, [IsPGAutomorphism], SUM_FLAGS, function( auto ) return PGInverse( auto ); end ); autpgrp-1.12.0/gap/autos.gd0000644000000000000000000000400515202202400012403 0ustar00############################################################################# ## #W autos.gd AutPGrp package Bettina Eick ## ############################################################################# ## #C Choose functionality ## if not IsBound( InitAutGroup ) then InitAutGroup := false; fi; if not IsBound( CHOP_MULT ) then CHOP_MULT := true; fi; if not IsBound( NICE_STAB ) then NICE_STAB := true; fi; if not IsBound( REDU_OPER ) then REDU_OPER := false; fi; if not IsBound( USE_LABEL ) then USE_LABEL := false; fi; if not IsBound( CHECK ) then CHECK := false; fi; ############################################################################# ## #D Declarations for PGAutomorphisms ## DeclareRepresentation( "IsPGAutomorphismRep", IsGroupGeneralMappingByImages, ["base", "baseimgs", "pcgs", "pcgsimgs"] ); IsPGAutomorphism := IsMapping and IsPGAutomorphismRep; DeclareOperation( "PGAutomorphism", [ IsPGroup and IsFinite, IsList, IsList ] ); DeclareGlobalFunction( "AutomorphismGroupPGroup" ); DeclareGlobalFunction( "PcGroupAutPGroup" ); DeclareGlobalFunction( "ConvertHybridAutGroup" ); DeclareGlobalFunction( "PGOrbitStabilizer" ); DeclareGlobalFunction( "IdentityPGAutomorphism" ); DeclareGlobalFunction( "CountOrbitsGL" ); DeclareGlobalFunction( "NumberOfPClass2PGroups" ); DeclareGlobalFunction( "NumberOfClass2LieAlgebras" ); DeclareOperation( "PGMult", [IsObject, IsObject] ); DeclareOperation( "PGInverse",[IsObject] ); DeclareOperation( "PGPower",[IsInt, IsObject] ); DeclareOperation( "PGMultList", [IsList] ); ############################################################################ ## #V for external applications ## DeclareGlobalFunction( "ImageAutPGroup" ); DeclareGlobalFunction( "InnerAutGroupPGroup" ); DeclareGlobalFunction( "ConvertAutGroup" ); DeclareGlobalFunction( "InduceAutGroup" ); DeclareGlobalFunction( "LinearActionAutGrp" ); DeclareGlobalFunction( "AddInfoCover" ); DeclareGlobalFunction( "InitAutomorphismGroupOver" ); autpgrp-1.12.0/gap/autos.gi0000644000000000000000000003005615202202400012415 0ustar00############################################################################# ## #W autos.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F LinearActionPGAut(

, , ) ## BindGlobal( "LinearActionPGAut", function( P, M, aut ) local p, gensP, pcgsM, gensG, defn, imgs, mat, d; # set up p := PrimePGroup( P ); gensP := GeneratorsOfGroup( P ); pcgsM := Pcgs( M ); gensG := DifferenceLists( gensP, pcgsM ); # compute matrix defn := P!.definitions; imgs := List( aut!.baseimgs, x -> MappedPcElement( x, aut!.pcgs, gensG )); for d in defn do if not IsNegRat( d ) then Add( imgs, SubstituteDef( d, imgs, p ) ); fi; od; imgs := imgs{List( pcgsM, x -> Position( gensP, x ) )}; # two cases - the first for efficiency if imgs = pcgsM then aut!.mat := 1; else mat := List(imgs, x -> ExponentsOfPcElement(pcgsM, x)*One(M!.field)); mat := ImmutableMatrix( M!.field, mat ); aut!.mat := mat; fi; end ); ############################################################################# ## #F LinearActionAutGrp( ,

, ) ## InstallGlobalFunction( LinearActionAutGrp, function( A, P, M ) local aut; # add information for aut in A.glAutos do LinearActionPGAut( P, M, aut ); od; for aut in A.agAutos do LinearActionPGAut( P, M, aut ); od; A.field := M!.field; A.prime := PrimePGroup( P ); A.one!.mat := 1; end); ############################################################################# ## #F CentralAutos( , ) ## BindGlobal( "CentralAutos", function( G, N ) local base, pcgs, cent, b, i, imgs, aut; base := Pcgs(N); pcgs := Pcgs(G); cent := []; for b in base do for i in [1..RankPGroup(G)] do imgs := ShallowCopy( pcgs ); imgs[i] := imgs[i] * b; aut := PGAutomorphism( G, pcgs, imgs ); Add( cent, aut ); od; od; return cent; end ); ############################################################################# ## #F InduceAuto( , ) ## BindGlobal( "InduceAuto", function( F, aut ) local pcgsF, baseF, imgsG, imgsF, hom; pcgsF := Pcgs( F ); baseF := pcgsF{[1..RankPGroup(F)]}; imgsG := aut!.baseimgs; imgsF := List( imgsG, x -> MappedPcElement( x, aut!.pcgs, pcgsF ) ); if CHECK then hom := GroupHomomorphismByImages( F, F, baseF, imgsF ); if not IsGroupHomomorphism( hom ) then Error("no hom"); elif not IsBijective( hom ) then Error("no bijection"); fi; fi; return PGAutomorphism( F, baseF, imgsF ); end ); ############################################################################# ## #F InduceAutGroup( , ,

, , ) ## InstallGlobalFunction( InduceAutGroup, function( A, Q, P, M, U ) local p, r, F, s, t, pcgsF, pcgsL, L, B, central; # set up p := PrimePGroup( P ); r := RankPGroup( P ); # create factor F := Range( EpimorphismQuotientSystem(Q) ); SetIsPGroup( F, true ); SetPrimePGroup( F, p ); SetRankPGroup( F, r ); pcgsF := Pcgs(F); # get definitions for F F!.definitions := RewriteDef( pcgsF, Q!.definitions, p ); # get p-centre of F s := Length( Pcgs(P) ) - Length( Pcgs(M) ); t := s + Length( Pcgs(M) ) - Length( Pcgs(U) ); pcgsL := InducedPcgsByPcSequenceNC( pcgsF, pcgsF{[s+1..t]} ); L := SubgroupByPcgs( F, pcgsL ); # induce autos B := rec(); B.glAutos := List( A.glAutos, x -> InduceAuto( F, x ) ); B.agAutos := List( A.agAutos, x -> InduceAuto( F, x ) ); central := CentralAutos( F, L ); Append( B.agAutos, central ); # add information B.glOrder := A.glOrder; B.agOrder := Concatenation( A.agOrder, List( central, x -> p ) ); B.group := F; B.one := IdentityPGAutomorphism( F ); B.size := B.glOrder * Product( B.agOrder ); # if possible add projective operation if IsBound( A.glOper ) then B.glOper := A.glOper; fi; # and return return B; end); ############################################################################# ## #F ConvertAuto( , ) ## BindGlobal( "ConvertAuto", function( aut, iso ) local G, pcgs, imgs, auto; G := Source( iso ); pcgs := SpecialPcgs( G ); imgs := List( pcgs, x -> ImagesRepresentative( iso, x ) ); imgs := List( imgs, x -> ImagesRepresentative( aut, x ) ); imgs := List( imgs, x -> PreImagesRepresentative( iso, x ) ); if not CHECK then auto := GroupHomomorphismByImagesNC(G, G, pcgs, imgs ); SetIsBijective( auto, true ); else auto := GroupHomomorphismByImages( G, G, AsList(pcgs), imgs ); if not IsGroupHomomorphism( auto ) then Error("automorphism is no homomorphism"); elif not IsBijective( auto ) then Error("automorphism is not bijective"); fi; fi; return auto; end ); ############################################################################# ## #F ConvertAutGroup ( , ) ## InstallGlobalFunction( ConvertAutGroup, function( A, G ) local r, gens, imgs, iso, C; r := RankPGroup( G ); gens := SpecialPcgs( G ){[1..r]}; imgs := Pcgs( A.group ){[1..r]}; if not CHECK then iso := GroupHomomorphismByImagesNC( G, A.group, gens, imgs ); SetIsBijective( iso, true ); else iso := GroupHomomorphismByImages( G, A.group, gens, imgs ); if not IsGroupHomomorphism( iso ) then Error("isomorphism is no homomorphism"); elif not IsBijective( iso ) then Error("isomorphism is not bijective"); fi; fi; C := rec(); C.glAutos := List( A.glAutos, x -> ConvertAuto( x, iso ) ); C.glOrder := A.glOrder; C.agAutos := List( A.agAutos, x -> ConvertAuto( x, iso ) ); C.agOrder := A.agOrder; C.one := IdentityMapping( G ); C.group := G; C.size := A.size; # if possible add projective operation if IsBound( A.glOper ) then C.glOper := A.glOper; fi; return C; end); ############################################################################# ## #F AddInfoCover( Q, P, M, U ) ## InstallGlobalFunction( AddInfoCover, function( Q, P, M, U ) local r, p, f, fam, gensP, pcgsP, gensM, pcgsM, gensU, pcgsU, pos, def; r := Q!.RanksOfDescendingSeries; p := Q!.prime; f := Q!.field; fam := FamilyPcgs( P ); # info for P gensP := GeneratorsOfGroup( P ); pcgsP := InducedPcgsByPcSequenceNC( fam, gensP ); SetPcgs( P, pcgsP ); SetRankPGroup( P, r[1] ); SetPrimePGroup( P, p ); # info for M gensM := GeneratorsOfGroup( M ); pcgsM := InducedPcgsByPcSequenceNC( fam, gensM ); SetPcgs( M, pcgsM ); SetPrimePGroup( M, p ); M!.field := f; # info for U gensU := GeneratorsOfGroup( U ); pcgsU := InducedPcgsByGeneratorsNC( fam, gensU ); SetPcgs( U, pcgsU ); # get definitions of M pos := List( pcgsP, x -> Position( fam, x ) ); def := Q!.definitions{pos}; P!.definitions := RewriteDef( pcgsP, def, p ); end); ############################################################################# ## #F AutomorphismGroupPGroup( , ) . . . .automorphisms in hybird form ## InstallGlobalFunction( AutomorphismGroupPGroup, function( arg ) local p, r, G, pcgs, first, n, str, A, F, Q, i, s, t, P, N, M, U, B, baseU, baseN, epi, f; # catch the trivial case G := arg[1]; p := PrimePGroup( G ); r := RankPGroup( G ); if Size( G ) = 1 then return Group( [], IdentityMapping(G) ); fi; # choose a initialisation if Length( arg ) = 1 then if IsHomoCyclic( G ) then InitAutGroup := InitAutomorphismGroupFull; elif (p^r - 1)/(p - 1) < 30000 then InitAutGroup := InitAutomorphismGroupOver; else InitAutGroup := InitAutomorphismGroupChar; fi; elif Length( arg ) = 2 and IsString(arg[2]) then if arg[2] = "Full" then InitAutGroup := InitAutomorphismGroupFull; elif arg[2] = "Over" then InitAutGroup := InitAutomorphismGroupOver; elif arg[2] = "Char" then InitAutGroup := InitAutomorphismGroupChar; else Error("invalid initialisation"); fi; elif Length( arg ) = 2 and IsBool(arg[2]) and arg[2] = true then str := Interrupt("choose initialisation (Over/Char/Full)"); if str = "Over" then InitAutGroup := InitAutomorphismGroupOver; elif str = "Char" then InitAutGroup := InitAutomorphismGroupChar; elif str = "Full" then InitAutGroup := InitAutomorphismGroupFull; else Error("invalid initialisation"); fi; fi; # choose flags CHOP_MULT := true; NICE_STAB := true; USE_LABEL := false; # compute special pcgs pcgs := SpecialPcgs( G ); first := LGFirst( SpecialPcgs(G) ); p := PrimePGroup( G ); n := Length(pcgs); r := RankPGroup( G ); f := GF(p); # init automorphism group - compute Aut(G/G_1) Info( InfoAutGrp, 1, "step 1: ",p,"^", first[2]-1, " -- init automorphisms "); A := InitAutGroup( G ); # loop over remaining steps F := Range( IsomorphismFpGroupByPcgs( pcgs, "f" ) ); Q := PQuotient( F, p, 1 ); for i in [2..Length(first)-1] do # print info s := first[i]; t := first[i+1]; Info( InfoAutGrp, 1, "step ",i,": ",p,"^", t-s, " -- aut grp has size ", A.size ); # the cover Info( InfoAutGrp, 2, " computing cover"); P := PCover( Q ); M := PMultiplicator( Q, P ); N := Nucleus( Q, P ); U := AllowableSubgroup( Q, P ); AddInfoCover( Q, P, M, U ); # induced action of A on M Info( InfoAutGrp, 2, " computing matrix action"); LinearActionAutGrp( A, P, M ); # compute stabilizer Info( InfoAutGrp, 2, " computing stabilizer of U"); baseN := GeneratorsOfGroup(N); baseU := GeneratorsOfGroup(U); baseN := List(baseN, x -> ExponentsOfPcElement(Pcgs(M), x)) * One(f); baseU := List(baseU, x -> ExponentsOfPcElement(Pcgs(M), x)) * One(f); baseU := EcheloniseMat( baseU ); PGOrbitStabilizer( A, baseU, baseN, false ); # next step of p-quotient IncorporateCentralRelations( Q ); RenumberHighestWeightGenerators( Q ); # induce to next factor Info( InfoAutGrp, 2, " induce autos and add central autos"); A := InduceAutGroup( A, Q, P, M, U ); od; # now get a real automorphism group Info( InfoAutGrp, 1, "final step: convert"); return ConvertAutGroup( A, G ); end ); ############################################################################# ## #M ConvertHybridAutGroup( ) ## InstallGlobalFunction( ConvertHybridAutGroup, function( A ) local B, pcgs; B := Group( Concatenation( A.glAutos, A.agAutos ), A.one ); SetSize( B, A.glOrder * Product( A.agOrder ) ); if Length( A.glAutos ) = 0 then SetIsSolvableGroup( B, true ); pcgs := PcgsByPcSequenceNC( FamilyObj( A.one ), A.agAutos ); SetRelativeOrders( pcgs, A.agOrder ); SetOneOfPcgs( pcgs, A.one ); SetGeneralizedPcgs( B, pcgs ); fi; SetIsGroupOfAutomorphisms( B, true ); return B; end ); ############################################################################# ## #M AutomorphismGroup ## InstallMethod( AutomorphismGroup, "for finite p-groups", true, [IsPGroup and IsFinite and CanEasilyComputePcgs], 0, function( G ) local A; # the trivial case is a problem if Size( G ) = 1 then return Group( [], IdentityMapping(G) ); fi; # compute A := AutomorphismGroupPGroup( G ); # translate and return A:=ConvertHybridAutGroup( A ); SetIsAutomorphismGroup(A,true); if IsFinite(G) then SetIsGroupOfAutomorphismsFiniteGroup(A,true); fi; return A; end ); autpgrp-1.12.0/gap/countcl.gi0000644000000000000000000002647715202202400012745 0ustar00############################################################################# ## #W countcl.gi Bettina Eick ## #W Let GL(n,p) act linearly on some space V. The function in this file can #W be used to count the number of orbits of subspaces of dimension k arising #W in this action. ## #W As an application, the function in this file can be used to count the #W number of p-groups of p-class 2 with n generators or the number of #W Lie algebras over GF(p) with n generators. ## ############################################################################# ## #F InducedActionFactor( mats, fac, low ) ## BindGlobal( "InducedActionFactor@", function( mats, fac, low ) local sml, upp, d, i, b, t; sml := List( mats, x -> [] ); upp := Concatenation( fac, low ); d := Length( fac ); for i in [1..Length(mats)] do for b in fac do t := SolutionMat( upp, b*mats[i] ){[1..d]}; Add( sml[i], t ); od; od; return sml; end ); ############################################################################# ## #F SpinUpCyclic( v, mat, d ) ## BindGlobal( "SpinUpCyclic", function( v, mat, d ) local b, i; b := [v]; for i in [1..d-1] do b[i+1] := b[i] * mat; od; TriangulizeMat(b); return b; end ); ############################################################################# ## #F JordanBlockLengths( mat, F ) ## BindGlobal( "JordanBlockLengths", function( g, F ) local chr, min, fc, fm, ns, V, W, U, i, f, e, h, k, v, t; # char poly chr := CharacteristicPolynomial(g); fc := Collected(Factors(chr)); # min poly min := MinimalPolynomial(F, g); fm := Factors(min); # set up ns := List( fc, x -> List( [1..x[2]], y -> 0 ) ); V := g^0; # loop for i in [1..Length(fc)] do # determine V_i f := fc[i][1]; e := Length(Filtered(fm, x -> x = f)); h := Value( f^e, g ); W := TriangulizedNullspaceMat(h); k := InducedActionFactor@( [g], W, [] )[1]; # split cyclic while not IsBool(k) do ns[i][e] := ns[i][e] + 1; if Length(k) = e * Degree(f) then k := false; else h := Value( f^(e-1), k ); v := First( k^0, x -> x*h <> 0*x ); W := SpinUpCyclic( v, k, e*Degree(f) ); U := BaseSteinitzVectors(k^0, W).factorspace; k := InducedActionFactor@( [k], U, W )[1]; t := Collected(Factors(MinimalPolynomial( F, k ))); e := t[1][2]; fi; od; od; return rec( blks := ns, degs := List(fc, x -> Degree(x[1]))); end ); ############################################################################# ## #F Two small help functions ## BindGlobal( "WeightedSum", function(vec) return Sum(List( [1..Length(vec)], x -> x * vec[x] )); end ); BindGlobal( "PartialSums", function(vec) return List( [1..Length(vec)], x -> Sum(vec{[x..Length(vec)]})); end ); ############################################################################# ## #F Deltas( blls, degs, k ) ## BindGlobal( "Deltas", function( ni, d, k ) local ds, i, r, new, del, s, j, m; ds := [[]]; i := 1; r := Length(d); while i <= r do new := []; for del in ds do s := Sum(List([1..i-1], x -> del[x]*d[x])); if i < r then m := Minimum( ni[i], QuoInt(k-s, d[i]) ); for j in [0..m] do Add( new, Concatenation( del, [j] ) ); od; fi; if i = r then j := (k-s) / d[i]; if IsInt(j) and j <= ni[i] then Add( new, Concatenation( del, [j] ) ); fi; fi; od; ds := new; i := i + 1; od; return ds; end ); ############################################################################# ## #F Gammas( deli, vis ) ## BindGlobal( "Gammas", function( d, v ) local par, i; par := Partitions( d ); for i in [1..Length(par)] do if Length(par[i]) <= Length(v) and ForAll( [1..Length(par[i])], x -> par[i][x] <= v[x] ) then par[i] := Concatenation(par[i], 0*[1..Length(v)-Length(par[i])]); else par[i] := false; fi; od; return Filtered(par, x -> not IsBool(x)); end ); ############################################################################# ## #F PChoose( n, k, q ) . . . . . . . . . .number of k-dim subspaces in GF(q)^n ## BindGlobal( "PChoose", function( n, k, q ) local qn, qd, i, size; if n / 2 < k then k:= n - k; fi; size:= 1; qn:= q^n; qd:= q; for i in [ 1 .. k ] do size:= size * ( qn - 1 ) / ( qd - 1 ); qn:= qn / q; qd:= qd * q; od; return size; #return Size(Subspaces(GF(q)^n, k)); end ); ############################################################################# ## #F Feval(a, b, q ) . . . . . . . . . . . . . . . . . . .evaluate f on a, b, q ## BindGlobal( "Feval", function( a, b, q ) local l, s, i; Add( b, 0 ); l := Length(a); s := 1; for i in [1..l] do s := s * q^(b[i+1]*(a[i]-b[i])); s := s * PChoose(a[i]-b[i+1], b[i]-b[i+1], q); od; return s; end ); ############################################################################# ## #F FixedPointsByDecom( blocks, degs, p, k ) ## BindGlobal( "FixedPointsByDecom", function( blks, degs, p, k ) local r, ni, ds, vi, t, td, tg, delta, gamma, i, gs; # set up r := Length(degs); # compute deltas ni := List( blks, WeightedSum ); ds := Deltas( ni, degs, k ); # precompute vij vi := List( blks, PartialSums ); # add up t := 0; for delta in ds do td := 1; for i in [1..r] do gs := Gammas( delta[i], vi[i] ); tg := 0; for gamma in gs do tg := tg + Feval( vi[i], gamma, p^degs[i] ); od; td := td * tg; od; t := t + td; od; # that's it return t; end ); ############################################################################ ## #F CountOrbitsGL( n, p, k, Action ) . . . . . . count the orbits of GL(n,p) ## ## Action defines the action and k the dimension of the subspaces to count. ## InstallGlobalFunction( CountOrbitsGL, function( n, p, k, Action ) local G, cls, jor, len, fix, cl, g, h, J, t, j, l; # set up G := GL(n,p); cls := ConjugacyClasses(G); # catch input if IsBool(k) then l := n*(n-1)/2-1; fi; # infos jor := []; len := []; fix := []; for cl in cls do # get g and action g := Representative(cl); h := Action(g, GF(p)); Info(InfoAutGrp, 1, "start class of order ", Order(g)); # compute jordan blocks n_{ij} J := JordanBlockLengths(h, GF(p)); # check if known j := Position( jor, J ); if IsBool( j ) then Info( InfoAutGrp, 2, " Degree: ",J.degs," -- Jordan ", J.blks); if IsBool(k) then t := List([1..l], x -> FixedPointsByDecom(J.blks,J.degs,p,x)); else t := FixedPointsByDecom( J.blks, J.degs, p, k ); fi; Add( jor, J ); Add( len, Size(cl) ); Add( fix, t ); Info( InfoAutGrp, 2, " class yields ",t); else len[j] := len[j] + Size(cl); fi; od; return Sum( List( [1..Length(jor)], i -> len[i]*fix[i] ) ) / Size(G); end ); ############################################################################# ## #F Some Actions ## BindGlobal( "WedgeAction", function( mat, f ) return WedgeGModule(GModuleByMats([mat], f)).generators[1]; end ); BindGlobal( "TensorAction", function( mat, f ) return KroneckerProduct( mat, mat ); end ); BindGlobal( "WedgePlusChar2Action", function( mat, f ) local d, e, M, i, j, l, k, a, b; # set up d := Length(mat); e := (d^2 - d)/2; M := NullMat(e+d,e+d); # wedge part for i in [2..d] do for j in [1..i-1] do a := (i-1)*(i-2)/2+j; for k in [2..d] do for l in [1..k-1] do b := (k-1)*(k-2)/2+l; M[a][b] := mat[i][k]*mat[j][l] - mat[i][l]*mat[j][k]; od; od; od; od; # power part for i in [1..d] do a := e+i; for k in [2..d] do for l in [1..k-1] do b := (k-1)*(k-2)/2+l; M[a][b] := mat[i][k]*mat[i][l]; od; od; for j in [1..d] do b := e+j; M[a][b] := mat[i][j]; od; od; return M * One(f); end ); BindGlobal( "WedgePlusCharPAction", function( mat, f ) return DirectSumMat( WedgeAction(mat,f), mat ); end ); BindGlobal( "WedgePlusAction", function( mat, f ) if Characteristic(f) = 2 then return WedgePlusChar2Action(mat, f); else return WedgePlusCharPAction(mat, f); fi; end ); ############################################################################ ## #F NumberOfPClass2PGroups( n, p, [k] ) ## InstallGlobalFunction( NumberOfPClass2PGroups, function( arg ) local n, p, l, k, c; n := arg[1]; p := arg[2]; l := n*(n+1)/2; if Length(arg) = 3 then if arg[3] > l then return 0; fi; if arg[3] = l then return 1; fi; if arg[3] = 0 then return 0; fi; return CountOrbitsGL( arg[1], arg[2], arg[3], WedgePlusAction ); elif Length(arg) = 2 then c := []; c[l] := 1; for k in [1..l-1] do if not IsBound( c[k] ) then c[k] := CountOrbitsGL( n, arg[2], k, WedgePlusAction ); c[l-k] := c[k]; fi; od; return c; fi; Error("wrong input"); end ); ############################################################################ ## #F NumberOfClass2LieAlgebras( n, p, [k] ) ## InstallGlobalFunction( NumberOfClass2LieAlgebras, function( arg ) local n, k, c, m; if Length(arg) = 3 then return CountOrbitsGL( arg[1], arg[2], arg[3], WedgeAction ); elif Length(arg) = 2 then n := arg[1]; m := n*(n-1)/2; c := []; c[m] := 1; for k in [1..m-1] do if not IsBound( c[k] ) then c[k] := CountOrbitsGL( n, arg[2], k, WedgeAction ); c[m-k] := c[k]; fi; od; return c; fi; Error("wrong input"); end ); ############################################################################ ## #F NumberOfClass2AssociativeAlgebras( n, p, [k] ) ## BindGlobal( "NumberOfClass2AssocAlgebras", function( arg ) local n, k, c, m; if Length(arg) = 3 then return CountOrbitsGL( arg[1], arg[2], arg[3], TensorAction ); elif Length(arg) = 2 then n := arg[1]; m := n*(n-1)/2; c := []; c[m] := 1; for k in [1..m-1] do if not IsBound( c[k] ) then c[k] := CountOrbitsGL( n, arg[2], k, TensorAction ); c[m-k] := c[k]; fi; od; return c; fi; Error("wrong input"); end ); BindGlobal( "NumberOfClass2AssocAlgebrasByDim", function( d, p ) local r, n, c; r := 2; for n in [2..d-1] do c := NumberOfClass2AssocAlgebras( n, p, d-n ); r := r+c; od; return r; end ); autpgrp-1.12.0/gap/general.gi0000644000000000000000000001055615202202400012702 0ustar00############################################################################# ## #W general.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F Interrupt ## BindGlobal( "Interrupt", function(text) local str, ans; Print("\n",text); Print(": \c"); str := InputTextUser(); ans := ReadLine(str); ans := ans{[1..Length(ans)-1]}; Print("\n"); return ans; end ); ############################################################################# ## #F RewriteDef( pcgs, defn, p ) ## BindGlobal( "RewriteDef", function( pcgs, defn, p ) local words, i, d, e, w; words := []; for i in [1..Length(defn)] do d := defn[i]; if IsNegRat( d ) then Add( words, d ); elif IsInt( d ) then w := pcgs[d]^p; e := ExponentsOfPcElement( pcgs, w ); e[i] := 0; Add( words, [d, e] ); elif IsList( d ) then w := Comm( pcgs[d[1]], pcgs[d[2]] ); e := ExponentsOfPcElement( pcgs, w ); e[i] := 0; Add( words, [d, e] ); fi; od; return words; end ); ############################################################################# ## #F SubstituteDef( def, gens, p ) ## BindGlobal( "SubstituteDef", function( def, gens, p ) local i, e; # definition part if IsInt( def ) then return gens[-def]; elif IsInt( def[1] ) then e := gens[def[1]]^p; else e := Comm( gens[def[1][1]], gens[def[1][2]] ); fi; # tail part for i in [1..Length(def[2])] do if def[2][i] <> 0 then e := gens[i]^-def[2][i] * e; fi; od; return e; end ); ############################################################################# ## #F InducedPcgsByBasis( pcgs, basis ) ## BindGlobal( "InducedPcgsByBasis", function( pcgs, basis ) local pcgsN, pcgsM, seq, pcs; pcgsN := NumeratorOfModuloPcgs( pcgs ); pcgsM := DenominatorOfModuloPcgs( pcgs ); seq := List( basis, b -> PcElementByExponents( pcgs, b ) ); pcs := InducedPcgsByPcSequenceAndGenerators( pcgsN, pcgsM, seq ); return InducedPcgsByPcSequenceNC( pcgsN, pcs ); end ); ############################################################################# ## #F IsHomoCyclic( G ) ## BindGlobal( "IsHomoCyclic", function( G ) return IsAbelian(G) and Length(Set(AbelianInvariants(G))) = 1; end ); ############################################################################# ## #F FrattiniQuotientPGroup( G ) ## BindGlobal( "FrattiniQuotientPGroup", function( G ) local spec, firs, frat, H; spec := SpecialPcgs(G); firs := LGFirst(spec); frat := InducedPcgsByPcSequenceNC( spec, spec{[firs[2]..Length(spec)]}); H := GroupByPcgs( spec mod frat ); SetIsPGroup(H, true ); SetPrimePGroup( H, PrimePGroup(G) ); SetRankPGroup( H, firs[2] - 1 ); H!.definitions := List( [1..firs[2]-1], x -> -x ); return H; end ); ############################################################################# ## #F InitGlAutos( H, mats ) ## BindGlobal( "InitGlAutos", function( H, mats ) local pcgs; pcgs := Pcgs(H); return List( mats, x -> PGAutomorphism( H, pcgs, List( x, y -> PcElementByExponents( pcgs, y) ) ) ); end ); ############################################################################# ## #F InitAgAutos( H, p ) ## BindGlobal( "InitAgAutos", function( H, p ) local pcgs, auts, alpha, fac, i, imgs; if p <> 2 then pcgs := Pcgs(H); auts := []; alpha := PrimitiveRoot( GF(p) ); fac := Factors( p - 1 ); for i in [1..Length(fac)] do imgs := List( pcgs, x-> x^IntFFE( alpha ) ); Add( auts, PGAutomorphism( H, pcgs, imgs ) ); alpha := alpha ^ fac[i]; od; return rec( auts := auts, rels := fac ); else return rec( auts := [], rels := [] ); fi; end ); ############################################################################# ## #F EcheloniseMat( mat ) ## BindGlobal( "EcheloniseMat", function( mat ) local ech, tmp, i; if Length(mat) = 0 then return mat; fi; ech := SemiEchelonMat( mat ); tmp := []; for i in [1..Length(ech.heads)] do if ech.heads[i] <> 0 then Add( tmp, ech.vectors[ech.heads[i]] ); fi; od; return tmp; end); autpgrp-1.12.0/gap/hybrstab.gi0000644000000000000000000002433415202202400013102 0ustar00############################################################################# ## #W hybridst.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F CollectToWord( list ) ## BindGlobal( "CollectToWord", function( list ) local coll, t, i; coll := []; t := [list[1], 1]; for i in [2..Length(list)] do if t[1] <> list[i] then Add( coll, t ); t := [list[i], 1]; else t[2] := t[2] + 1; fi; od; Add( coll, t ); return coll; end ); ############################################################################# ## #F TransformPG( get, list, id ) . . . . . . . . . . . .convert get to element ## BindGlobal( "TransformPG", function( get, list, id ) local coll, res, i; # catch the special case if Length( get ) = 0 then return id; fi; # otherwise compute coll := CollectToWord( get ); if coll[1][1] > 0 then res := [PGPower( coll[1][2],list[coll[1][1]] )]; else res := [PGPower( coll[1][2], PGInverse(list[-coll[1][1]]))]; fi; for i in [2..Length(coll)] do if coll[i][1] > 0 then Add( res, PGPower(coll[i][2],list[coll[i][1]] ) ); else Add( res, PGPower(coll[i][2], PGInverse(list[-coll[i][1]]) ) ); fi; od; return PGMultList( res ); end ); ############################################################################# ## #F Transform( get, list, id ) . . . . . . . . . . . . .convert get to element ## BindGlobal( "Transform", function( get, list, id ) local res, i; if Length( get ) = 0 then return id; fi; if get[1] > 0 then res := list[get[1]]; else res := list[-get[1]]^-1; fi; for i in [2..Length( get )] do if get[i] > 0 then res := res * list[get[i]]; else res := res * list[-get[i]]^-1; fi; od; return res; end ); ############################################################################# ## #F ReduceGet( ords, get ) . . . . . . . . . . . . . . .reduce get with orders ## BindGlobal( "ReduceGet", function( ords, get ) local found, i, j, o; found := true; while found do # first reduce by inverses i := 1; found := false; while i <= Length( get ) - 1 do if not IsBool( get[i+1] ) and get[i] = - get[i+1] then get[i] := false; get[i+1] := false; found := true; i := i + 1; fi; i := i + 1; od; # now reduce by orders i := 1; while i <= Length( get ) do if not IsBool( get[i] ) then if get[i] > 0 then o := ords[ get[i] ]; else o := ords[ -get[i] ]; fi; if i+o-1 <= Length( get ) and ForAll( get{[i..i+o-1]}, x -> x = get[i] ) then for j in [i..i+o-1] do get[j] := false; od; found := true; i := i + o - 1; fi; fi; i := i + 1; od; od; return Filtered( get, x -> not IsBool( x ) ); end ); ############################################################################# ## #F OSTransversalInverse( j, trans, trels, id ) ## BindGlobal( "OSTransversalInverse", function( j, trans, trels, id ) local l, g, s, p, t; if j = 1 then return id; fi; l := Product( trels ); j := j - 1; g := id; for s in Reversed( [1..Length( trans )] ) do p := trels[s]; l := l/p; t := QuoInt( j, l ); j := RemInt( j, l ); if t > 0 then g := PGMult( g, PGInverse(trans[s])^t ); fi; od; return g; end ); ############################################################################# ## #F PcgsOrbitStabilizer( A, oper, pt, fpt, info ) ## BindGlobal( "PcgsOrbitStabilizer", function( A, oper, pt, fpt, info ) local pcgs, rels, stabl, srels, trans, trels, orbit, i, y, j, p, l, s, k, t, h, g, dict; pcgs := A.agAutos; rels := A.agOrder; # catch trivial case if Length( pcgs ) = 0 then return rec( stabl := pcgs, srels := rels, orbit := [pt], trans := [], trels := [] ); fi; # initialise orbit, stabiliser and transversal stabl := []; srels := []; trans := []; trels := []; orbit := [pt]; dict := NewDictionary( pt, true ); AddDictionary( dict, pt, 1 ); # Start constructing orbit. i := Length( pcgs ); while i >= 1 do if oper[i] = 1 then Add( stabl, pcgs[i] ); Add( srels, rels[i] ); else y := fpt( pt, oper[i], info ); j := LookupDictionary( dict, y ); if IsBool( j ) then # enlarge transversal Add( trans, pcgs[i] ); Add( trels, rels[i] ); # enlarge orbit p := rels[i]; l := Length( orbit ); orbit[p*l] := true; s := 0; for k in [ 1 .. p - 1 ] do t := s + l; for h in [ 1 .. l ] do orbit[h + t] := fpt( orbit[h + s], oper[i], info ); AddDictionary( dict, orbit[h + t], h + t ); od; s := t; od; else # enlarge stabilizer if j > 1 then g := OSTransversalInverse(j, trans, trels, A.one); Add( stabl, PGMult( pcgs[i], g ) ); else Add( stabl, pcgs[i] ); fi; Add( srels, rels[i] ); fi; fi; i := i - 1; od; return rec( stabl := Reversed( stabl ), srels := Reversed( srels ), orbit := orbit, trans := trans, trels := trels ); end ); ############################################################################# ## #F BlockOrbitStabilizer( B, oper, os, fpt, info ) ## BindGlobal( "BlockOrbitStabilizer", function( B, oper, os, fpt, info ) local bl, l, li, orbit, trans, stabl, pstab, mats, auts, ords, pers, k, pt, i, y, j, new, get, aut, g, per, s, dict, r, stabGrp; # the block and limit for orbit length bl := os.orbit; l := Length( bl ); li := B.glOrder / Factors( B.glOrder )[1]; # set up orbit, transversal and stab orbit := [ bl ]; dict := NewDictionary( bl[1], true ); for j in [1..l] do AddDictionary( dict, bl[j], [1,j] ); od; trans := [ [] ]; stabl := []; pstab := []; # get acting elements auts := B.glAutos; ords := List( auts, Order ); stabGrp := Group( () ); if IsBound( B.glOper ) then pers := B.glOper; fi; # loop k := 1; while k <= Length( orbit ) do if k mod 10000 = 0 then Info( InfoAutGrp, 5, " orbit pos ", k, " of ",Length(orbit)); fi; pt := orbit[k][1]; for i in [ 1..Length(oper) ] do # compute the image of a point y := fpt( pt, oper[i], info ); j := LookupDictionary( dict, y ); if IsBool( j ) then # enlarge orbit and transversal new := List( [1..l], x -> true ); for s in [1..l] do new[s] := fpt( orbit[k][s], oper[i], info ); AddDictionary( dict, new[s], [Length(orbit)+1, s] ); od; Add( orbit, new ); get := Concatenation( trans[k], [i] ); Add( trans, get ); else # enlarge stabilizer get := Concatenation(trans[k], [i], Reversed(-trans[j[1]])); get := ReduceGet( ords, get ); aut := TransformPG( get, auts, B.one ); # reduce from block-stab to point-stab if j[2] > 1 then g := OSTransversalInverse(j[2], os.trans, os.trels, B.one); aut := PGMult( aut, g ); fi; # add permutations if known if IsBound( B.glOper ) then g := Transform( get, pers, () ); if not g in stabGrp then stabGrp := ClosureGroup( stabGrp, g ); Add( pstab, g ); Add( stabl, aut ); fi; else Add( stabl, aut ); fi; fi; od; if Length( orbit ) * Size(stabGrp) > li then # orbit is going to be the whole domain, and the # stabilizer cannot get any larger, so we are done return rec( stabl := stabl, pstab := pstab, length := B.glOrder / Size(stabGrp) ); else k := k + 1; fi; od; return rec( stabl := stabl, pstab := pstab, length := Length(orbit) ); end ); ############################################################################# ## #F PGHybridOrbitStabilizer( A, glMats, agMats, pt, oper, info ) ## BindGlobal( "PGHybridOrbitStabilizer", function( A, glMats, agMats, pt, oper, info ) local os, OS, B; # compute ag orbit stabilizier if Length( glMats ) = 0 and Length( agMats ) = 0 then return; fi; os := PcgsOrbitStabilizer( A, agMats, pt, oper, info ); Info( InfoAutGrp, 4, " ag-orbit -- length ",Length(os.orbit)); # add info to A A.agAutos := os.stabl; A.agOrder := os.srels; # compute block orbit and stabiliser if Length( glMats ) = 0 then return; fi; OS := BlockOrbitStabilizer( A, glMats, os, oper, info ); Info( InfoAutGrp, 4, " gl-orbit -- length ", OS.length, " -- gens ",Length(OS.stabl)); # set up new aut grp A.glAutos := OS.stabl; A.glOrder := A.glOrder / OS.length; Assert(1,IsInt(A.glOrder)); if IsBound( A.glOper ) then A.glOper := OS.pstab; fi; # nice the glAutos if necessary if NICE_STAB and OS.length > 1 then NiceHybridGroup( A ); fi; end ); autpgrp-1.12.0/gap/initmat.gi0000644000000000000000000001413215202202400012724 0ustar00############################################################################# ## #W initmat.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F TwoStepCentralizersByLcs( G ) ## ## The two-step-centralizers of lower p-central series of G. ## BindGlobal( "TwoStepCentralizersByLcs", function( G ) local pcgs, first, p, field, list, i, f, m, n, max, pcgsN, pcgsM, pcgsH, gensL, gensC, pcgsR, new; # set up pcgs := SpecialPcgs( G ); first := LGFirst( pcgs ); p := PrimePGroup( G ); field := GF(p); list := []; max := Length(pcgs); # run through lower p-central series for i in [3..Length(first)] do f := first[i-2]; m := first[i-1]; n := first[i]; pcgsN := InducedPcgsByPcSequenceNC( pcgs, pcgs{[f..max]} ); pcgsM := InducedPcgsByPcSequenceNC( pcgs, pcgs{[m..max]} ); pcgsH := InducedPcgsByPcSequenceNC( pcgs, pcgs{[n..max]} ); gensL := pcgsN mod pcgsM; gensC := pcgs mod pcgsM; pcgsR := pcgsM mod pcgsH; new := NextStepCentralizer( gensL, gensC, pcgsR, field ); Append(new, pcgsM ); #new := InducedPcgsByPcSequenceNC( pcgs, new ); new := InducedPcgsByGeneratorsNC( pcgs, new ); Add( list, SubgroupByPcgs( G, new ) ); od; return list; end ); ############################################################################# ## #F OmegaSubgroupsByLcs( G ) ## ## The preimages of Omega-subgroups of G_i for all factors G_i of the lower ## p-central series of G. ## BindGlobal( "OmegaSubgroupsByLcs", function( G ) local pcgs, first, p, field, list, max, i, pcgsN, N, hom, F, ser, specF; # catch the trivial case p := PrimePGroup( G ); pcgs := SpecialPcgs( G ); if ForAll( pcgs, x -> Order(x) = p ) then return []; fi; # set up first := LGFirst( pcgs ); field := GF(p); list := []; max := Length(pcgs); # run through lower p-central series for i in [2..Length(first)] do pcgsN := InducedPcgsByPcSequenceNC( pcgs, pcgs{[first[i]..max]} ); N := SubgroupByPcgs( G, pcgsN ); hom := NaturalHomomorphismByNormalSubgroupNC( G, N ); F := Image( hom ); specF := SpecialPcgs(F); if ForAny( specF, x -> Order(x) > p ) and Size(F) < 10000 then ser := OmegaSeries( F ); ser := List( ser, x -> PreImage( hom, x ) ); Append( list, ser ); fi; od; return list; end ); ############################################################################# ## #F MaxSubIntersections(G) ## BindGlobal( "MaxSubIntersections", function(G) local max, fp, fpset, fpbin, i, j; max := MaximalSubgroups(G); fp := List(max, IdGroup); fpset := Set(fp); fpbin := List(fpset, x -> []); for i in [1..Length(max)] do j := Position(fpset, fp[i]); Add(fpbin[j], max[i]); od; return List(fpbin, Intersection); end ); ############################################################################# ## #F PGCharSubgroups( G ) ## BindGlobal( "PGCharSubgroups", function(G) local cent, omega; cent := TwoStepCentralizersByLcs( G ); omega := OmegaSubgroupsByLcs( G ); if Size(G) <= 512 then return Union(cent, omega, MaxSubIntersections(G)); else return Union( cent, omega ); fi; end ); ############################################################################# ## #F FrattiniQuotientBase( , ) ## BindGlobal( "FrattiniQuotientBase", function( spec, U ) local r, frat, pcgs, subU, base; r := LGFirst(spec)[2]; frat := InducedPcgsByPcSequenceNC( spec, spec{[r..Length(spec)]} ); pcgs := spec mod frat; subU := Filtered(InducedPcgs(spec, U), x -> DepthOfPcElement(spec,x) ExponentsOfPcElement( pcgs, x ) ); return base; end ); ############################################################################# ## #F InitAutomorphismGroupChar( G ) ## BindGlobal( "InitAutomorphismGroupChar", function( G ) local r, p, chars, bases, S, H, A, z, spec, kern; Info( InfoAutGrp, 2, " init automorphism group : Char "); # set up r := RankPGroup( G ); p := PrimePGroup( G ); z := One(GF(p)); spec := SpecialPcgs( G ); # compute characteristic subgroups Info( InfoAutGrp, 3, " compute characteristic subgroups "); chars := PGCharSubgroups( G ); bases := List( chars, x -> FrattiniQuotientBase( spec, x ) ) * z; # compute the matrixgroup stabilising all subspaces in chain Info( InfoAutGrp, 3, " compute stabilizer "); S := StabilizingMatrixGroup( bases, r, p ); # the Frattini Quotient H := FrattiniQuotientPGroup( G ); kern := InitAgAutos( H, p ); # the aut group A := rec( ); A.glAutos := InitGlAutos( H, GeneratorsOfGroup(S) ); A.glOrder := Size(S) / Product( kern.rels ); A.glOper := GeneratorsOfGroup(S); Assert(1,IsInt(A.glOrder)); A.agAutos := kern.auts; A.agOrder := kern.rels; A.one := IdentityPGAutomorphism( H ); A.group := H; A.size := A.glOrder * Product( A.agOrder ); # try to construct perm rep NiceInitGroup( A, true ); return A; end ); ############################################################################# ## #F InitAutomorphismGroupFull( G ) ## BindGlobal( "InitAutomorphismGroupFull", function( G ) local r, p, S, H, A, kern; Info( InfoAutGrp, 2, " init automorphism group : Full "); # set up r := RankPGroup( G ); p := PrimePGroup( G ); S := GL(r, p); H := FrattiniQuotientPGroup( G ); kern := InitAgAutos( H, p ); # the aut group A := rec( ); A.glAutos := InitGlAutos( H, GeneratorsOfGroup(S) ); A.glOrder := Size(S) / Product( kern.rels ); A.glOper := GeneratorsOfGroup( S ); Assert(1,IsInt(A.glOrder)); A.agAutos := kern.auts; A.agOrder := kern.rels; A.one := IdentityPGAutomorphism( H ); A.group := H; A.size := A.glOrder * Product( A.agOrder ); # try to compute perm rep NiceInitGroup( A, false ); return A; end ); autpgrp-1.12.0/gap/initperm.gi0000644000000000000000000001257715202202400013121 0ustar00############################################################################# ## #W initperm.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F Fingerprint( G, U ) ## BindGlobal( "FingerprintSmall@", function( G, U ) return Flat( [IdGroup( U ), Size(CommutatorSubgroup( G, U )) ]); end ); BindGlobal( "FingerprintMedium@", function( G, U ) local ranks, invs, comm, all, cls, fus, new; # some general stuff ranks := LGFirst( SpecialPcgs( U ) ); invs := AbelianInvariants( Centre(U) ); comm := Size( CommutatorSubgroup( G, U ) ); # use conjugacy classes all := Orbits( G, AsList(U) ); all := List( all, x -> Set(x)); cls := List( all, x -> Order(x[1]) ); Sort( cls ); return Concatenation( ranks, invs, [comm], cls ); end ); BindGlobal( "FingerprintLarge@", function( G, U ) return LGFirst( SpecialPcgs(U) ); end ); BindGlobal( "FingerprintHuge@", function( G, U ) return List( DerivedSeries(U), Size ); end ); BindGlobal( "PGFingerprint", function ( G, U ) if Size( U ) <= 256 and IsRecord( ID_AVAILABLE( Size(U) ) ) then return FingerprintSmall@( G, U ); elif Size( U ) <= 1000 then return FingerprintMedium@( G, U ); elif Size( U ) <= 2^21 then return FingerprintLarge@( G, U ); else return FingerprintHuge@( G, U ); fi; end ); # FIXME: DualBasis is not used in autpgrp; it is however # used in the sophus package. The next sophus release won't use # it anymore. But for now we keep this function here to not # break existing sophus releases... DualBasis := function( base ) local M; M := NullspaceMat( TransposedMat( base )); M := List( M, ShallowCopy ); TriangulizeMat( M ); return M; end; ############################################################################# ## #F PartitionMinimalOvergrps ( G, pcgs, norm ) ## BindGlobal( "PartitionMinimalOvergrps", function( G, pcgs, norm ) local min, done, part, i, tup, pos; Info( InfoAutGrp, 3, " computing partition "); done := []; part := []; for i in [1..Length(norm)] do Info( InfoAutGrp, 4, " start ",i); min := InducedPcgsByBasis( pcgs, [norm[i]] ); tup := PGFingerprint( G, SubgroupByPcgs( G, min ) ); pos := Position( done, tup ); if IsBool( pos ) then Add( part, [i] ); Add( done, tup ); else Add( part[pos], i ); fi; od; Sort( part, function( x, y ) return Length(x) < Length(y); end ); return part; end ); ############################################################################# ## #F PartitionStabilizer ( A, part, norm ) ## BindGlobal( "PartitionStabilizer", function( A, part, norm ) local iso, P, sub, gens, n, q; Info( InfoAutGrp, 3, " computing stabilizer of ", part); iso := ActionHomomorphism( A, norm, OnLines, "surjective" ); P := Image( iso ); # transfer size info n := DimensionOfMatrixGroup(A); q := Size(FieldOfMatrixGroup(A)); if HasIsNaturalGL(A) and IsNaturalGL(A) then SetSize(P,Size(A)/(q-1)); elif HasIsNaturalSL(A) and IsNaturalSL(A) then SetSize(P,Size(A)/Gcd(n,q-1)); fi; # loop for sub in part{[1..Length(part)-1]} do if Length( sub ) = 1 then P := Stabilizer( P, sub[1], OnPoints ); Info( InfoAutGrp, 3, " found stabilizer of size ", Size(P)); else P := Stabilizer( P, sub, OnSets ); Info( InfoAutGrp, 3, " found stabilizer of size ", Size(P)); fi; od; gens := SmallGeneratingSet( P ); # return return rec( perm := gens, mats := List( gens, x -> PreImagesRepresentative(iso,x) ), size := Size(P) ); end ); ############################################################################# ## #F AutoOfMat( mat, H ) ## BindGlobal( "AutoOfMat", function( mat, H ) local img, aut, pcgs; pcgs := Pcgs(H); img := List( mat, x -> PcElementByExponentsNC(pcgs, x) ); aut := PGAutomorphism( H, pcgs, img ); return aut; end ); ############################################################################# ## #F InitAutomorphismGroupOver( G ) ## InstallGlobalFunction( InitAutomorphismGroupOver, function( G ) local r, p, pcgsG, pcgsN, pcgs, base, V, norm, part, stab, H, kern, A; Info( InfoAutGrp, 2, " initialize automorphism group: Over "); # set up r := RankPGroup( G ); p := PrimePGroup( G ); # pgcs'se pcgsG := SpecialPcgs(G); pcgsN := InducedPcgsByPcSequenceNC( pcgsG, pcgsG{[r+1..Length(pcgsG)]} ); pcgs := pcgsG mod pcgsN; # get partition stabilizer base := IdentityMat( r, GF(p) ); V := GF(p)^r; norm := NormedRowVectors( V ); part := PartitionMinimalOvergrps( G, pcgs, norm ); stab := PartitionStabilizer( GL( r, p ), part, norm ); # get quotient H := FrattiniQuotientPGroup( G ); kern := InitAgAutos( H, p ); # create aut grp A := rec(); A.glAutos := List( stab.mats, x -> AutoOfMat( x, H ) ); A.glOrder := stab.size; A.glOper := ShallowCopy( stab.perm ); A.agAutos := kern.auts; A.agOrder := kern.rels; A.one := IdentityPGAutomorphism(H); A.group := H; A.size := A.glOrder * Product( A.agOrder ); # try to construct solvable normal subgroup NiceInitGroup( A, true ); return A; end); autpgrp-1.12.0/gap/matrix.gi0000644000000000000000000000641315202202400012566 0ustar00############################################################################# ## #W matrix.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F ChainByCollection( bases, full ) ## BindGlobal( "ChainByCollection", function( bases, full ) local chain, base, tmp, i, V, W, int; chain := [ full, [] ]; for base in bases do tmp := []; for i in [2..Length(chain)] do V := chain[i-1]; W := chain[i]; if Length( V ) > Length( W ) + 1 then int := SumIntersectionMat( base, V )[2]; int := SumIntersectionMat( int, W )[1]; if Length( int ) < Length( V ) and Length( int ) > Length( W ) then AddSet( tmp, int ); fi; fi; od; Append( chain, tmp ); Sort( chain, function( x, y ) return Length(x)>Length(y); end ); od; return chain; end ); ############################################################################# ## #F StabilizingMatrixGroup( list of bases ) . . . . . compute stabilizer in GL ## BindGlobal( "StabilizingMatrixGroup", function( bases, d, p ) local full, chain, mats, l, size, i, j, n, mat, G, rel, field; # general set up full := IdentityMat( d, GF(p) ); chain := ChainByCollection( bases, full ); field := GF(p); # loop over chain mats := []; l := 0; size := 1; for i in [2..Length(chain)] do n := Length( chain[i-1] ) - Length( chain[i] ); for j in [1..n] do size := size * (p^(d-l) - p^(d-l-j)); od; # Construct the generators. if p = 2 then if n >= 2 then mat := IdentityMat(d, field); mat[l+1][l+n] := One( field ); mat[l+1][l+1] := Zero( field ); for j in [ 2 .. n ] do mat[l+j][l+j-1] := One( field ); mat[l+j][l+j] := Zero( field ); od; Add( mats, mat ); mat := IdentityMat(d, field); mat[l+1][l+2] := One( field ); Add( mats, mat ); fi; else mat := IdentityMat(d, field); mat[l+1][l+1] := PrimitiveRoot( field ); Add( mats, mat ); if n >= 2 then mat := IdentityMat(d, field); mat[l+1][l+1] := -One( field ); mat[l+1][l+n] := One( field ); for j in [ 2 .. n ] do mat[l+j][l+j-1] := -One( field ); mat[l+j][l+j] := Zero( field ); od; Add( mats, mat ); fi; fi; l := l + n; if l < d then mat := IdentityMat(d, field); mat[l][l+1] := One( field ); Add( mats, mat ); fi; od; # change basis of mats rel := List( [2..Length(chain)], i -> BaseSteinitzVectors( chain[i-1], chain[i] ).factorspace ); rel := Concatenation( rel ); mats := List( mats, x -> rel^-1 * x * rel ); G := Group( mats, IdentityMat( d, field ) ); SetSize( G, size ); return G; end ); autpgrp-1.12.0/gap/matrstab.gi0000644000000000000000000001604015202202400013074 0ustar00############################################################################# ## #W matrstab.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F LabelOfBasis( base, info ) . . . . . . . . . . . . . . . . . label to basis ## BindGlobal( "LabelOfBasis", function( base, info ) local pt, j, i; # compute pt pt := List( [1..info.l], x -> 0 ); for j in [1..info.l] do for i in [1..info.d] do if base[j][i] <> Zero( info.field ) then pt[j] := pt[j] + IntFFE( base[j][i] ) * info.power[i]; fi; od; od; # create label return pt; end ); ############################################################################# ## #F CoeffsInt( int, d, power ) ## BindGlobal( "CoeffsInt", function( int, d, power) local i, exp; i := d; exp := List( power, y -> 0 ); while int <> 0 do exp[i] := QuoInt (int, power[i]); int := RemInt (int, power[i]); i := i - 1; od; return exp; end ); ############################################################################# ## #F BasisOfLabel( lab, info ) . . . . . . . . . . . . . . . . . basis of label ## BindGlobal( "BasisOfLabel", function ( lab, info ) return List( [1..info.l], x -> CoeffsInt( lab[x], info.d, info.power ) ); end ); ############################################################################# ## #F OnLabel( lab, mat, info ) . . . . . . . . . . . . . . . .operation on label ## BindGlobal( "OnLabel", function( lab, mat, info ) local v, w; v := BasisOfLabel( lab, info ); w := v * mat; TriangulizeMat( w ); return LabelOfBasis( w, info ); end ); ############################################################################# ## #F OnBasis( base, mat, info ) . . . . . . . . . . . . . . .operation on basis ## BindGlobal( "OnBasis", function( base, mat, info ) base := base * mat; if not IsMutable(base) then base:=ShallowCopy(base); fi; TriangulizeMat( base ); return base; end ); ############################################################################# ## #F InducedActionByHom( hom, mat ) ## BindGlobal( "InducedActionByHom", function( hom, mat ) local ind, baseI, baseS, b, e, f, g; # catch special case for efficiency if mat = 1 then return mat; fi; # otherwise compute ind := []; baseI := Basis( ImagesSource( hom ) ); baseS := Basis( Source( hom ) ); for b in baseI do e := PreImagesRepresentative( hom, b ); f := e * mat; g := ImagesRepresentative( hom, f ); Add( ind, Coefficients( baseI, g ) ); od; # again catch a special case for efficiency if ind = ind^0 then return 1; fi; # otherwise return return ImmutableMatrix( Characteristic(ind[1][1]), ind); end ); ############################################################################# ## #F ActionOnDual( mat ) ## BindGlobal( "ActionOnDual", function( mat ) local new; if mat = 1 then return 1; fi; new := TransposedMat( mat )^-1; return ImmutableMatrix( Characteristic(new[1][1]), new); end ); ############################################################################# ## #F PGMatrixOrbitStabilizer( A, V, W, R ) ## BindGlobal( "PGMatrixOrbitStabilizer", function( A, V, W, R ) local VS, WS, RS, hom, pt, glMats, agMats, lab, info, l, d, oper; # set up factor space VS := VectorSpace( A.field, V, "basis" ); WS := VectorSpace( A.field, W, Zero(VS), "basis" ); RS := VectorSpace( A.field, R, "basis" ); hom := NaturalHomomorphismBySubspaceOntoFullRowSpace( VS, WS ); pt := ShallowCopy( Basis( ImagesSet( hom, RS ) ) ); TriangulizeMat( pt ); # set up action glMats := List( A.glAutos, x -> InducedActionByHom( hom, x!.mat ) ); agMats := List( A.agAutos, x -> InducedActionByHom( hom, x!.mat ) ); # check if the dual is better if Length(pt) > Length(pt[1])/2 then pt := VectorSpace( A.field, pt, "basis" ); pt := OrthogonalSpaceInFullRowSpace( pt ); pt := ShallowCopy( Basis( pt ) ); TriangulizeMat( pt ); glMats := List( glMats, x -> ActionOnDual( x ) ); agMats := List( agMats, x -> ActionOnDual( x ) ); fi; # use labels - if desired if USE_LABEL then d := Length( pt[1] ); l := Length( pt ); info := rec( power := List( [1..d], x -> A.prime^(x-1) ), field := A.field, l := l, d := d ); lab := LabelOfBasis( pt, info ); PGHybridOrbitStabilizer( A, glMats, agMats, lab, OnLabel, info ); else pt := ImmutableMatrix( A.field, pt ); PGHybridOrbitStabilizer( A, glMats, agMats, pt, OnBasis, rec() ); fi; end ); ############################################################################# ## #F CheckStab(A, pt) ## BindGlobal( "CheckAgStab", function( A, pt ) local g, s, c; for g in A.agAutos do for s in pt do c := SolutionMat(pt, s*g!.mat); if IsBool(c) then return false; fi; od; od; return true; end ); BindGlobal( "CheckGlStab", function( A, pt ) local g, s, c; for g in A.glAutos do for s in pt do c := SolutionMat(pt, s*g!.mat); if IsBool(c) then return false; fi; od; od; return true; end ); ############################################################################# ## #F SumMat( mat1, mat2 ) ## BindGlobal( "SumMat@" ,function( mat1, mat2 ) local tmp; tmp := Concatenation( mat1, mat2 ); tmp := EcheloniseMat( tmp ); TriangulizeMat(tmp); return tmp; end ); ############################################################################# ## #F PGOrbitStabilizerBySeries( A, baseU, chop ) ## BindGlobal( "PGOrbitStabilizerBySeries", function( A, baseU, chop ) local s, U, T, i, S, R, j, V, W, B; s := Length(chop); U := baseU; # loop upwards over series T := []; for i in [1..s-1] do S := ShallowCopy(T); T := SumIntersectionMat( U, chop[i+1] )[2]; if Length( S ) < Length( T ) then # loop downwards over series R := chop[i+1]; for j in Reversed( [1..i] ) do V := ShallowCopy( R ); R := SumMat@( T, chop[j] ); if Length( R ) < Length( V ) then W := SumMat@( S, chop[j] ); if Length( R ) > Length( W ) then PGMatrixOrbitStabilizer( A, V, W, R ); if CHECK then if not CheckAgStab(A, R) then Error("ag stab wrong "); fi; if not CheckGlStab(A, R) then Error("gl stab wrong "); fi; fi; else Info( InfoAutGrp, 4, " skip trivial factor"); fi; fi; od; fi; od; end ); autpgrp-1.12.0/gap/nicestab.gi0000644000000000000000000001411615202202400013051 0ustar00############################################################################# ## #W nicestab.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F GLMatrix( aut ) ## BindGlobal( "GLMatrix", function( aut ) local G, r, pcgsG, pcgsN, pcgsF, mat; G := Source( aut ); r := RankPGroup( G ); pcgsG := Pcgs( G ); pcgsN := InducedPcgsByPcSequenceNC( pcgsG, pcgsG{[r+1..Length(pcgsG)]} ); pcgsF := pcgsG mod pcgsN; mat := List(pcgsF, x->ExponentsOfPcElement(pcgsF,ImagesRepresentative(aut,x))); return mat; end ); ############################################################################# ## #F TryPermOperation( A ) . . . . . . . . resets A.glOper to perms or nothing ## BindGlobal( "TryPermOperation", function( A ) local G, r, p, base, V, norm, f, M, iso, P; # if its too big, then don't try. G := A.group; r := RankPGroup( G ); p := PrimePGroup( G ); if (p^r - 1) / (p - 1) > 1100 then Unbind( A.glOper ); return; fi; Info( InfoAutGrp, 4, " compute perm rep"); # now we compute it base := IdentityMat( r, GF(p) ); V := GF(p)^r; norm := NormedRowVectors( V ); f := function( pt, a ) return NormedRowVector( pt * a ); end; M := Group( A.glOper, base ); iso := ActionHomomorphism( M, norm, f ); P := Image( iso ); # and get images A.glOper := GeneratorsOfGroup( P ); end ); ############################################################################# ## #F ReducePermOper( A ) ## BindGlobal( "ReducePermOper", function(A) local P, B, phom, gens, auts; Info( InfoAutGrp, 4, " reduce permutation operation"); # get perm group P := Group( A.glOper, ()); B := Group( A.glAutos ); SetSize( P, A.glOrder ); # mapping from A to permgroup P phom := GroupHomomorphismByImagesNC( B, P, A.glAutos, A.glOper ); gens := SmallGeneratingSet(P); gens := Filtered( gens, x -> Order(x) > 1 ); auts := List( gens, x -> PreImagesRepresentative( phom, x ) ); A.glAutos := auts; A.glOper := gens; Info( InfoAutGrp, 4, " factor has size ",A.glOrder," and ", Length(A.glAutos)," generators"); end ); ############################################################################# ## #F TrySolvableSubgroup( A ) ## BindGlobal( "TrySolvableSubgroup", function( A ) local P, B, N, pcgs, phom, nhom, G, auts, gens; Info( InfoAutGrp, 4, " try solvable normal subgroup"); # get perm group P := Group( A.glOper, ()); B := Group( A.glAutos ); SetSize( P, A.glOrder ); # mapping from A to permgroup P phom := GroupHomomorphismByImagesNC( B, P, A.glAutos, A.glOper ); # get normal subgroup N := SolvableRadical( P ); pcgs := Pcgs( N ); Info( InfoAutGrp, 4, " found pcgs of length ", Length(pcgs)); if Length(pcgs) = 0 then return; fi; # construct factor nhom := NaturalHomomorphismByNormalSubgroupNC( P, N ); G := ImagesSource( nhom ); # get ag part auts := List( pcgs, x -> PreImagesRepresentative( phom,x ) ); A.agAutos := Concatenation( auts, A.agAutos ); A.agOrder := Concatenation( RelativeOrders( pcgs ), A.agOrder ); # and the factor phom := phom * nhom; gens := SmallGeneratingSet( G ); gens := Filtered( gens, x -> Order(x) > 1 ); auts := List( gens, x -> PreImagesRepresentative( phom, x ) ); A.glAutos := auts; A.glOrder := Size( G ); A.glOper := gens; Info( InfoAutGrp, 4, " factor has size ",A.glOrder," and ", Length(A.glAutos)," generators"); end ); ############################################################################# ## #F NiceInitGroup( A, "init" ) . . . . . . . . flag indicates the init method ## ## try to compute a perm rep and, if successful, compute N. ## BindGlobal( "NiceInitGroup", function( A, flag ) Info( InfoAutGrp, 3, " nice init group"); # catch a trivial case if Length( A.glOper ) = 0 then Unbind( A.glOper ); return; fi; # try to compute perm-oper, if possible if IsMatrix( A.glOper[1] ) then TryPermOperation( A ); return; fi; # finally, if a perm oper is given, then try to enlarge agAutos if IsPerm( A.glOper[1] ) and flag then if REDU_OPER then ReducePermOper( A ); else TrySolvableSubgroup( A ); fi; fi; end ); ############################################################################# ## #F NiceHybridGroup( A ) ## BindGlobal( "NiceHybridGroup", function( A ) local mats, done, auts, i, mat, aut, fac, rels, e, f; # catch the trivial cases if Length( A.glAutos ) = 0 or A.glOrder = 1 then Unbind( A.glOper ); A.glautos := []; return; fi; # in case we have a perm rep if IsBound( A.glOper ) then Info( InfoAutGrp, 3, " nice stabilizer with perm rep"); if REDU_OPER then ReducePermOper(A); else TrySolvableSubgroup( A ); fi; return; fi; Info( InfoAutGrp, 3, " nice stabilizer with matr rep"); # otherwise mats := List( A.glAutos, x -> GLMatrix( x ) * One( A.field ) ); done := [mats[1]^0]; auts := []; for i in [1..Length(mats)] do if not mats[i] in done then Add( auts, A.glAutos[i] ); Add( done, mats[i] ); fi; od; # catch a special case if Length( auts ) = 1 then mat := done[2]; aut := auts[1]; fac := Factors( Order( mat ) ); rels := []; auts := []; e := 1; for f in fac do Add( auts, aut^e ); e := e * f; od; A.agAutos := Concatenation( auts, A.agAutos ); A.agOrder := Concatenation( fac, A.agOrder ); A.glAutos := []; A.glOrder := 1; else A.glAutos := auts; fi; Info( InfoAutGrp, 4, " factor has size ",A.glOrder," and"); Info( InfoAutGrp, 4, Length(A.glAutos)," generators"); end ); autpgrp-1.12.0/gap/orbstab.gi0000644000000000000000000000507115202202400012715 0ustar00############################################################################# ## #W orbstab.gi AutPGrp package Bettina Eick ## ############################################################################# ## #F BasesCompositionSeriesThrough( M, base ) ## BindGlobal( "BasesCompositionSeriesThrough", function( M, base ) local full, chop, indu, smll, i, facb; full := IdentityMat( M.dimension, M.field ); chop := [[]]; # chop N if Length( base ) > 0 then indu := MTX.InducedActionSubmoduleNB( M, base ); smll := MTX.BasesCompositionSeries( indu ); for i in [2..Length( smll )] do smll[i] := smll[i] * base; Add( chop, EcheloniseMat( smll[i] ) ); od; fi; # chop M/N if Length( base ) < Length( full ) then indu := MTX.InducedActionFactorModuleWithBasis( M, base ); facb := indu[2]; indu := indu[1]; smll := MTX.BasesCompositionSeries( indu ); for i in [2..Length( smll )] do smll[i] := smll[i] * facb; Append( smll[i], base ); Add( chop, EcheloniseMat( smll[i] ) ); od; fi; return chop; end ); ############################################################################# ## #F PGOrbitStabilizer( , , , ) ## InstallGlobalFunction( PGOrbitStabilizer, function( A, baseU, baseN, interrupt ) local u, n, l, baseM, str, glMats, agMats, mats, modu, chop; # set up and catch some trivial cases u := Length( baseU ); n := Length( baseN ); if u = 0 or ( A.glOrder = 1 and Length( A.agOrder ) = 0 ) then return; fi; l := Length( baseU[1] ); baseM := IdentityMat( l, A.field ); if l = u then return; fi; # print some info Info( InfoAutGrp, 3, " dim U = ",u, " dim N = ",n, " dim M = ",l ); # check interrupt if interrupt then str := Interrupt("chop M/N and N: (y/n)"); if str = "y" then CHOP_MULT := true; elif str = "n" then CHOP_MULT := false; else Print("not a valid argument"); return; fi; fi; # compute series glMats := List( A.glAutos, x -> x!.mat ); agMats := List( A.agAutos, x -> x!.mat ); mats := Filtered( Concatenation( glMats, agMats ), x -> x<>1 ); modu := GModuleByMats( mats, l, A.field ); chop := [[], baseN, baseM]; if CHOP_MULT then chop := BasesCompositionSeriesThrough( modu, chop[2] ); fi; PGOrbitStabilizerBySeries( A, baseU, chop ); end ); autpgrp-1.12.0/gap/pcpres.gi0000644000000000000000000001364415202202400012562 0ustar00############################################################################# ## #W autpgrp.gi Bettina Eick ## #W The input to the following function is the output of #W AutomorphismGroupPGroup( G ) of the autpgrp package. ## #W The function computes a pc presentation for the solvable subgroup #W of Aut(G) defined by the agAutos of the input. ## ############################################################################# ## #F PcgsInfoAutPGroup( A ) . . . . . . . . . . . . . . . set up info on layers ## BindGlobal( "PcgsInfoAutPGroup", function( A ) local G, d, p, f, spec, gens, layer, bases, i, auto, imgs, base, j, s, n, B, P, M, pcgs, e; # set up G := A.group; d := RankPGroup(G); p := PrimePGroup(G); f := GF(p); spec := SpecialPcgs( G ); gens := spec{[1..d]}; # catch layers and bases layer := []; bases := []; for i in [1..Length(A.agAutos)] do auto := A.agAutos[i]; imgs := List( gens, x -> ExponentsOfPcElement( spec, x^auto ) ); base := imgs{[1..d]}{[1..d]}; # if base <> idmat, then the FrattiniFactor is permuted if base <> IdentityMat(d) then layer[i] := 1; bases[i] := base * One(f); else for j in [1..d] do imgs[j][j] := 0; od; e := Minimum( List( imgs, PositionNonZero ) ); j := LGLayers( spec )[e]; s := LGFirst( spec )[j]; n := LGFirst( spec )[j+1]; base := imgs{[1..d]}{[s..n-1]}; layer[i] := j; bases[i] := Concatenation( base ) * One(f); fi; od; # set up B := rec(); B.agAutos := A.agAutos; B.agOrder := A.agOrder; B.bases := bases; B.layer := layer; B.group := G; B.field := f; B.spec := spec; B.gens := gens; B.rank := d; # the first layer is particularly nasty j := PositionNot( layer, 1 ); if j > 1 then M := Group( bases{[1..j-1]}, IdentityMat(d, f) ); B.agHomom := IsomorphismPermGroup(M); P := Image(B.agHomom); pcgs := List( bases{[1..j-1]}, x -> Image(B.agHomom,x) ); pcgs := PcgsByPcSequence( FamilyObj(One(P)), pcgs ); SetRelativeOrders( pcgs, B.agOrder ); B.agTopfc := pcgs; fi; return B; end ); ############################################################################# ## #F ExponentsAutPGroup( B, auto ) . . . . . . . . . . .compute exponent vector ## BindGlobal( "ExponentsAutPGroup", function( B, auto ) local exps, imgs, perm, news, tmpa, j, e, s, n, subs, base; exps := List( B.agAutos, x -> 0 ); imgs := List( B.gens, x -> ExponentsOfPcElement( B.spec, x^auto ) ); base := imgs{[1..B.rank]}{[1..B.rank]}; # if base <> idmat, then the FrattiniFactor is permuted if base <> IdentityMat(B.rank) then base := base * One(B.field); perm := Image( B.agHomom, base ); news := ExponentsOfPcElement( B.agTopfc, perm ); exps{[1..Length(news)]} := news; tmpa := MappedVector( news, B.agAutos{[1..Length(news)]} ); auto := tmpa^-1 * auto; fi; imgs := List( B.gens, x -> ExponentsOfPcElement( B.spec, x^auto ) ); for j in [1..B.rank] do imgs[j][j] := 0; od; e := Minimum( List( imgs, PositionNonZero ) ); while e <= Length( B.spec ) do # determine base for auto j := LGLayers( B.spec )[e]; s := LGFirst( B.spec )[j]; n := LGFirst( B.spec )[j+1]; base := Concatenation( imgs{[1..B.rank]}{[s..n-1]} ) * One(B.field); # determine bases for layer s := PositionSorted( B.layer, j ); n := PositionSorted( B.layer, j+1 ); subs := B.bases{[s..n-1]}; # solve and set exponents news := IntVecFFE( SolutionMat( subs, base ) ); exps{[s..n-1]} := news; # divide off and reset tmpa := MappedVector( news, B.agAutos{[s..n-1]} ); auto := tmpa^-1 * auto; imgs := List( B.gens, x -> ExponentsOfPcElement( B.spec, x^auto ) ); for j in [1..B.rank] do imgs[j][j] := 0; od; e := Minimum( List( imgs, PositionNonZero ) ); od; return exps; end ); ############################################################################# ## #F ImageAutPGroup( B, G, auto ) . . . . . . . . . . . . . . image in pc group ## InstallGlobalFunction( ImageAutPGroup, function( B, G, auto ) local exp; exp := ExponentsAutPGroup( B, auto ); return MappedVector( exp, GeneratorsOfGroup( G ) ); end); ############################################################################# ## #F PcGroupAutPGroup(A) . . . . . . . . . . . . . . . . . . . compute pc group ## InstallGlobalFunction( PcGroupAutPGroup, function( A ) local B, C, m, F, f, r, i, j, o, w, e; B := PcgsInfoAutPGroup( A ); m := Length( B.agAutos ); F := FreeGroup( m ); f := GeneratorsOfGroup(F); r := []; for i in [1..m] do for j in [i..m] do if i = j then o := B.agOrder[i]; w := B.agAutos[i]^o; e := ExponentsAutPGroup( B, w ); Add( r, f[i]^o / MappedVector( e, f ) ); else w := B.agAutos[j]^B.agAutos[i]; e := ExponentsAutPGroup( B, w ); Add( r, f[j]^f[i] / MappedVector( e, f ) ); fi; od; od; C := PcGroupFpGroup( F/r ); C!.autos := B.agAutos; C!.autrec := B; return C; end ); ############################################################################# ## #F InnerAutGroupPGroup( C ). . . . . . . . . . . . embed Inn(G) into pc group ## InstallGlobalFunction( InnerAutGroupPGroup, function( C ) local B, G, gens, auts, imgs, I; B := C!.autrec; G := B.group; gens := Pcgs(G); auts := List(gens, x -> PGAutomorphism(G, gens, List(gens, y->y^x))); imgs := List( auts, x -> ImageAutPGroup( B, C, x ) ); I := Subgroup(C, imgs ); return I; end); autpgrp-1.12.0/init.g0000644000000000000000000000031515202202400011300 0ustar00############################################################################# ## #W init.g AutPGrp package Bettina Eick ## ReadPackage( "autpgrp", "gap/autos.gd" ); autpgrp-1.12.0/makedoc.g0000644000000000000000000000123615202202400011743 0ustar00############################################################################# ## ## makedoc.g ## Copyright (C) 2025 James D. Mitchell ## ## Licensing information can be found in the README file of this package. ## ############################################################################# ## LoadPackage("AutoDoc"); includes := [ "intro.xml", "method.xml", "underl.xml", "influen.xml", "additional.xml", ]; # The actual call to AutoDoc AutoDoc(rec( autodoc := rec(scan_dirs := []), gapdoc := rec(files := []), extract_examples := true, scaffold := rec(includes := includes) )); QuitGap(); autpgrp-1.12.0/read.g0000644000000000000000000000151215202202400011250 0ustar00############################################################################# ## #W read.g AutPGrp package Bettina Eick ## if not IsBound(SolvableRadical) then # for backwards compatibility with GAP versions before 4.12 BindGlobal( "SolvableRadical", RadicalGroup ); fi; ReadPackage( "autpgrp", "gap/general.gi"); ReadPackage( "autpgrp", "gap/autoops.gi"); ReadPackage( "autpgrp", "gap/matrix.gi"); ReadPackage( "autpgrp", "gap/nicestab.gi"); ReadPackage( "autpgrp", "gap/initmat.gi"); ReadPackage( "autpgrp", "gap/initperm.gi"); ReadPackage( "autpgrp", "gap/hybrstab.gi"); ReadPackage( "autpgrp", "gap/matrstab.gi"); ReadPackage( "autpgrp", "gap/orbstab.gi"); ReadPackage( "autpgrp", "gap/autos.gi"); ReadPackage( "autpgrp", "gap/pcpres.gi"); ReadPackage( "autpgrp", "gap/countcl.gi"); autpgrp-1.12.0/tst/0000755000000000000000000000000015202202400011000 5ustar00autpgrp-1.12.0/tst/autpgrp01.tst0000644000000000000000000000356215202202400013365 0ustar00# AutPGrp, chapter 2 # # DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD! # # This file has been generated by AutoDoc. It contains examples extracted from # the package documentation. Each example is preceded by a comment which gives # the name of a GAPDoc XML file and a line range from which the example were # taken. Note that the XML file in turn may have been generated by AutoDoc # from some other input. # gap> START_TEST("autpgrp01.tst"); # doc/method.xml:36-48 gap> LoadPackage("autpgrp", false); true gap> G := PcGroupCode(619031068735, 32); # SmallGroup( 32, 15 ); gap> SetInfoLevel( InfoAutGrp, 1 ); gap> AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 32 #I final step: convert # doc/method.xml:56-84 gap> G := DihedralGroup( IsPermGroup, 2^5 );; gap> IsPGroup(G); true gap> CanEasilyComputePcgs(G); true gap> IsFinite(G); true gap> A := AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^1 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 8 #I step 4: 2^1 -- aut grp has size 32 #I final step: convert gap> A.1; Pcgs([ ( 2,16)( 3,15)( 4,14)( 5,13)( 6,12)( 7,11)( 8,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ]) -> [ ( 1, 2)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10), ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16), ( 1, 3, 5, 7, 9,11,13,15)( 2, 4, 6, 8,10,12,14,16), ( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,11,15)( 4, 8,12,16), ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16) ] gap> Order(A.1); 16 # gap> STOP_TEST("autpgrp01.tst", 1); autpgrp-1.12.0/tst/autpgrp02.tst0000644000000000000000000000507415202202400013366 0ustar00# AutPGrp, chapter 3 # # DO NOT EDIT THIS FILE - EDIT EXAMPLES IN THE SOURCE INSTEAD! # # This file has been generated by AutoDoc. It contains examples extracted from # the package documentation. Each example is preceded by a comment which gives # the name of a GAPDoc XML file and a line range from which the example were # taken. Note that the XML file in turn may have been generated by AutoDoc # from some other input. # gap> START_TEST("autpgrp02.tst"); # doc/underl.xml:90-122 gap> LoadPackage("autpgrp", false); true gap> H := PcGroupCode(297368117289422176, 729); # SmallGroup (729, 34); gap> A := AutomorphismGroupPGroup(H); #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert rec( agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ], agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [ ], glOper := [ ], glOrder := 1, group := , one := IdentityMapping( ), size := 52488 ) gap> ConvertHybridAutGroup( A ); # doc/underl.xml:132-144 gap> H := PcGroupCode(297368117289422176, 729);; # SmallGroup (729, 34); gap> A := AutomorphismGroupPGroup(H);; #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert gap> B := PcGroupAutPGroup( A ); gap> I := InnerAutGroupPGroup( B ); Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, of ... ]) # gap> STOP_TEST("autpgrp02.tst", 1); autpgrp-1.12.0/tst/manual.example-2.tst0000644000000000000000000000241615202202400014605 0ustar00gap> START_TEST(""); # gap> SetInfoLevel( InfoAutGrp, 1 ); # gap> G := PcGroupCode(619031068735, 32); # SmallGroup( 32, 15 ); gap> AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 32 #I final step: convert gap> G := DihedralGroup( IsPermGroup, 2^5 );; gap> IsPGroup(G); true gap> CanEasilyComputePcgs(G); true gap> IsFinite(G); true gap> A := AutomorphismGroup(G); #I step 1: 2^2 -- init automorphisms #I step 2: 2^1 -- aut grp has size 2 #I step 3: 2^1 -- aut grp has size 8 #I step 4: 2^1 -- aut grp has size 32 #I final step: convert gap> A.1; Pcgs([ (2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,3,5,7,9,11,13,15)(2,4,6,8,10, 12,14,16), (1,5,9,13)(2,6,10,14)(3,7,11,15)(4,8,12,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16) ]) -> [ (1,2)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16), (1,3,5,7,9,11,13,15)(2,4,6,8,10, 12,14,16), (1,5,9,13)(2,6,10,14)(3,7,11,15)(4,8,12,16), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16) ] gap> Order(A.1); 16 # gap> STOP_TEST( "" ,1); autpgrp-1.12.0/tst/manual.example-3.tst0000644000000000000000000000413215202202400014603 0ustar00gap> START_TEST(""); # gap> SetInfoLevel( InfoAutGrp, 1 ); # gap> H := PcGroupCode(297368117289422176, 729); # SmallGroup(729, 34); gap> A := AutomorphismGroupPGroup(H); #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert rec( agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2, f1, f3^2, f5^2, f4^2, f6^2 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2^2, f3^2*f5, f4^2*f6, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ], agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], glAutos := [ ], glOper := [ ], glOrder := 1, group := , one := IdentityMapping( ), size := 52488 ) gap> ConvertHybridAutGroup( A ); gap> H := PcGroupCode(297368117289422176, 729);; # SmallGroup(729, 34); gap> A := AutomorphismGroupPGroup(H);; #I step 1: 3^2 -- init automorphisms #I step 2: 3^1 -- aut grp has size 8 #I step 3: 3^2 -- aut grp has size 72 #I step 4: 3^1 -- aut grp has size 5832 #I final step: convert gap> B := PcGroupAutPGroup( A ); gap> I := InnerAutGroupPGroup( B ); Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, of ... ]) # gap> STOP_TEST( "" ,1); autpgrp-1.12.0/tst/more.tst0000644000000000000000000001315715202202400012505 0ustar00gap> START_TEST("more.tst"); gap> SetInfoLevel( InfoAutGrp, 1 ); # # # gap> G:= PcGroupCode(9219, 16); # SmallGroup(16,2); gap> StructureDescription(G); "C4 x C4" # gap> A := AutomorphismGroupPGroup(G);; #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 6 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 6 96 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Full");; #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 6 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 6 96 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Over");; #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 6 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2, 2, 3 ] 1 96 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Char");; #I step 1: 2^2 -- init automorphisms #I step 2: 2^2 -- aut grp has size 6 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 6 96 gap> ConvertHybridAutGroup( A ); # # # gap> G:= PcGroupCode(17734058326, 32); # SmallGroup(32,50); # gap> A := AutomorphismGroupPGroup(G);; #I step 1: 2^4 -- init automorphisms #I step 2: 2^1 -- aut grp has size 120 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 120 1920 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Full");; #I step 1: 2^4 -- init automorphisms #I step 2: 2^1 -- aut grp has size 20160 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 120 1920 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Over");; #I step 1: 2^4 -- init automorphisms #I step 2: 2^1 -- aut grp has size 120 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 120 1920 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Char");; #I step 1: 2^4 -- init automorphisms #I step 2: 2^1 -- aut grp has size 20160 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 2, 2 ] 120 1920 gap> ConvertHybridAutGroup( A ); # # # gap> G := PcGroupCode(16385, 3 ^ 5); # SmallGroup(3^5, 61); gap> StructureDescription(G); "C9 x C3 x C3 x C3" # gap> A := AutomorphismGroupPGroup(G);; #I step 1: 3^4 -- init automorphisms #I step 2: 3^1 -- aut grp has size 606528 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 3, 3, 3, 3, 3, 3, 3 ] 5616 49128768 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Full");; #I step 1: 3^4 -- init automorphisms #I step 2: 3^1 -- aut grp has size 24261120 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 3, 3, 3, 3, 3, 3, 3 ] 5616 49128768 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Over");; #I step 1: 3^4 -- init automorphisms #I step 2: 3^1 -- aut grp has size 606528 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 3, 3, 3, 3, 3, 3, 3 ] 5616 49128768 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Char");; #I step 1: 3^4 -- init automorphisms #I step 2: 3^1 -- aut grp has size 606528 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 3, 3, 3, 3 ] 303264 49128768 gap> ConvertHybridAutGroup( A ); # # # gap> G := PcGroupCode(16400, 11 ^ 5); # SmallGroup(11^5, 81); gap> StructureDescription(G); "C11 x C11 x ((C11 x C11) : C11)" # gap> A := AutomorphismGroupPGroup(G);; #I step 1: 11^4 -- init automorphisms #I step 2: 11^1 -- aut grp has size 2551047840000 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 5, 5, 11, 11, 11, 11, 11, 11, 11, 11 ] 1742400 37349891425440000 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Full");; #I step 1: 11^4 -- init automorphisms #I step 2: 11^1 -- aut grp has size 41393302251840000 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 5, 11, 11, 11, 11 ] 255104784000 37349891425440000 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Over");; #I step 1: 11^4 -- init automorphisms #I step 2: 11^1 -- aut grp has size 2551047840000 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 2, 5, 5, 11, 11, 11, 11, 11, 11, 11, 11 ] 1742400 37349891425440000 gap> ConvertHybridAutGroup( A ); # gap> A := AutomorphismGroupPGroup(G, "Char");; #I step 1: 11^4 -- init automorphisms #I step 2: 11^1 -- aut grp has size 2551047840000 #I final step: convert gap> SortedList(A.agOrder); A.glOrder; A.size; [ 2, 5, 11, 11, 11, 11 ] 255104784000 37349891425440000 gap> ConvertHybridAutGroup( A ); # gap> STOP_TEST( "more.tst" ,1); autpgrp-1.12.0/tst/number.tst0000644000000000000000000000220415202202400013022 0ustar00gap> START_TEST("number.tst"); gap> SetInfoLevel( InfoAutGrp, 0 ); # gap> NumberOfPClass2PGroups(4, 2); [ 6, 54, 604, 3566, 6709, 3566, 604, 54, 6, 1 ] gap> List([0..11], i -> NumberOfPClass2PGroups(4, 2, i)); [ 0, 6, 54, 604, 3566, 6709, 3566, 604, 54, 6, 1, 0 ] # gap> NumberOfPClass2PGroups(4, 3); [ 6, 60, 1361, 23361, 66667, 23361, 1361, 60, 6, 1 ] gap> List([0..11], i -> NumberOfPClass2PGroups(4, 3, i)); [ 0, 6, 60, 1361, 23361, 66667, 23361, 1361, 60, 6, 1, 0 ] # gap> NumberOfClass2LieAlgebras(4, 2); [ 2, 4, 6, 4, 2, 1 ] gap> List([0..7], i -> NumberOfClass2LieAlgebras(4, 2, i)); [ 1, 2, 4, 6, 4, 2, 1, 0 ] # gap> NumberOfClass2LieAlgebras(4, 3); [ 2, 4, 6, 4, 2, 1 ] gap> List([0..7], i -> NumberOfClass2LieAlgebras(4, 3, i)); [ 1, 2, 4, 6, 4, 2, 1, 0 ] # gap> NumberOfClass2AssocAlgebras(4, 2); [ 27, 37269, 83200199, 37269, 27, 1 ] gap> List([1..3], i -> NumberOfClass2AssocAlgebras(4, 2, i)); [ 27, 37269, 83200199 ] # gap> NumberOfClass2AssocAlgebras(4, 3); [ 46, 3205858, 585449616953, 3205858, 46, 1 ] gap> List([1..3], i -> NumberOfClass2AssocAlgebras(4, 3, i)); [ 46, 3205858, 585449616953 ] # gap> STOP_TEST( "number.tst" ,1); autpgrp-1.12.0/tst/testall.g0000644000000000000000000000026615202202400012624 0ustar00LoadPackage("autpgrp"); TestDirectory(DirectoriesPackageLibrary("autpgrp", "tst"), rec(exitGAP := true, testOptions := rec(compareFunction := "uptowhitespace"))); FORCE_QUIT_GAP(1);