%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A anupq.tex GAP documentation Eamonn O'Brien %A & Frank Celler %A & Benedikt Rothe %% %A @(#)$Id: anupq.tex,v 3.12 1994/06/10 02:39:12 vfelsch Rel $ %% %Y Copyright 1990-1992, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %H $Log: anupq.tex,v $ %H Revision 3.12 1994/06/10 02:39:12 vfelsch %H updated examples %H %H Revision 3.11 1994/05/19 13:47:15 sam %H fixed 'StandardPresentation' %H %H Revision 3.10 1994/04/27 09:01:08 fceller %H added 'PqHomomorphism' %H added new parameter "SubList" %H %H Revision 3.9 1993/07/27 11:32:31 fceller %H changed examples to new free group / relators style %H %H Revision 3.8 1993/06/28 10:01:52 fceller %H new ANU pq version %H %H Revision 3.6 1993/02/15 15:15:16 felsch %H examples fixed %H %H Revision 3.5 1993/02/01 11:40:52 fceller %H fixed an example %H %H Revision 3.4 1993/01/05 11:40:05 fceller %H some more fixes, changed "SetupFile" to return 'true' %H %H Revision 3.3 1993/01/04 11:01:01 fceller %H fixed some misspellings %H %H Revision 3.2 1992/12/29 15:57:49 fceller %H Initial GAP 3.2 revision %H %% \Chapter{ANU Pq} The ANU $p$-quotient program (pq) may be called from {\GAP}. Using this program, {\GAP} provides access to the following\:\ the $p$-quotient algorithm; the $p$-group generation algorithm; a standard presentation algorithm; an algorithm to compute the automorphism group of a $p$-group. The following section describes the function 'Pq', which gives access to the $p$-quotient algorithm. The next section describes the function 'PqDescendants', which gives access to the $p$-group generation algorithm. The next sections describe functions for saving results to file (see "PqList" and "SavePqList"). The next section describes the function 'StandardPresentation' which gives access to the standard presentation algorithm and to the algorithm used to compute the automorphism group of a $p$-group. The last sections describes the function 'IsIsomorphicPGroup' which implements an isomorphism test for $p$-groups using the standard presentation algorithm. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Pq} 'Pq( , ... )' Let be a finitely presented group. Then 'Pq' returns the desired $p$-quotient of as an ag group. The following parameters or parameter pairs are supported. \"Prime\",

: \\ Compute the $p$-quotient for the prime

. \"ClassBound\", : \\ The $p$-quotient computed has lower exponent-$p$ class at most . \"Exponent\", : \\ The $p$-quotient computed has exponent . By default, no exponent law is enforced. \"Metabelian\": \\ The largest metabelian $p$-quotient is constructed. \"Verbose\": \\ The runtime-information generated by the ANU pq is displayed. By default, pq works silently. \"OutputLevel\", : \\ The runtime-information generated by the ANU pq is displayed at output level , which must be a integer from 0 to 3. This parameter implies \"Verbose\". \"SetupFile\", : \\ Do not run the ANU pq, just construct the input file and store it in the file . In this case 'true' is returned. Alternatively, you can pass 'Pq' a record as a parameter, which contains as entries some (or all) of the above mentioned. Those parameters which do not occur in the record are set to their default values. See also "PqHomomorphism". | gap> RequirePackage("anupq"); gap> f2 := FreeGroup( 2, "f2" ); Group( f2.1, f2.2 ) gap> Pq( f2, rec( Prime := 2, ClassBound := 3 ) ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8, G.9, G.10 ) gap> g := f2 / [ f2.1^4, f2.2^4 ];; gap> Pq( g, rec( Prime := 2, ClassBound := 3 ) ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8 ) gap> Pq( g, "Prime", 2, "ClassBound", 3, "Exponent", 4 ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7 ) gap> g := f2 / [ f2.1^25, Comm(Comm(f2.2,f2.1),f2.1), f2.2^5 ];; gap> Pq( g, "Prime", 5, "Metabelian", "ClassBound", 5 ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7 ) | This function requires the package \"anupq\"\ (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PqHomomorphism} 'PqHomomorphism( , )' Let be a $p$-quotient of $F$ computed using 'Pq'. If is a list of images of '$F$.generators' under an automorphism of $F$, 'PqHomomorphism' will return the corresponding automorphism of . | gap> F := FreeGroup (2, "F"); Group( F.1, F.2 ) gap> G := Pq (F, "Prime", 5, "Class", 2); Group( G.1, G.2, G.3, G.4, G.5 ) gap> PqHomomorphism (G, [F.2, F.1]); GroupHomomorphismByImages( Group( G.1, G.2, G.3, G.4, G.5 ), Group( G.1, G.2, G.3, G.4, G.5 ), [ G.1, G.2, G.3, G.4, G.5 ], [ G.2, G.1, G.3^4, G.5, G.4 ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PqDescendants} 'PqDescendants( , ... )' Let be an ag group of prime power order with a consistent power-commutator presentation (see "IsConsistent"). 'PqDescendants' returns a list of descendants of . If does *not* have p-class 1, then a list of automorphisms of must be bound to the record component '.automorphisms' such that '.automorphisms' together with the inner automorphisms of generate the automorphism group of . One method which may be used to obtain such a generating set for the automorphism group is to call 'StandardPresentation'. The record returned has a generating set for the automorphism group of stored as a component (see "StandardPresentation"). The following optional parameters or parameter pairs are supported. \"ClassBound\", : \\ 'PqDescendants' generates only descendants with lower exponent-$p$ class at most . The default value is the exponent-$p$ class of plus one. \"OrderBound\", : \\ 'PqDescendants' generates only descendants of size at most $p^$. Note that you cannot set both \"OrderBound\"\ and \"StepSize\". \"StepSize\", : \\ Let be a positive integer. 'PqDescendants' generates only those immediate descendants which are $p^$ bigger than their parent group. \"StepSize\", : \\ Let be a list of positive integers such that the sum of the length of and the exponent-$p$ class of is equal to the class bound \"ClassBound\". Then describes the step size for each additional class. \"AgAutomorphisms\": \\ The automorphisms stored in '.automorphisms' are a PAG-generating sequence for the automorphism group of supplied in *reverse* order. \"RankInitialSegmentSubgroups\", : \\ Set the rank of the initial segment subgroup chosen to be . By default, this has value 0. \"SpaceEfficient\": \\ The ANU pq performs calculations more slowly but with greater space efficiency. This flag is frequently necessary for groups of large Frattini quotient rank. The space saving occurs because only one permutation is stored at any one time. This option is only available in conjunction with the \"AgAutomorphisms\"\ flag. \"AllDescendants\": \\ By default, only capable descendants are constructed. If this flag is set, compute all descendants. \"Exponent\", : \\ Construct only descendants with exponent . Default is no exponent law. \"Metabelian\": \\ Construct only metabelian descendants. \"SubList\", : \\ Let $L$ be the list of descendants generated. If list is supplied, 'PqDescendants' returns 'Sublist( $L$, )'. If an integer is supplied, 'PqDescendants' returns $L[]$. \"Verbose\": \\ The runtime-information generated by the ANU pq is displayed. By default, pq works silently. \"SetupFile\", : \\ Do not run the ANU pq, just construct the input file and store it in the file . In this case 'true' is returned. \"TmpDir\",

: \\ 'PqDescendants' stores intermediate results in temporary files; the location of these files is determined by the value selected by 'TmpName'. If your default temporary directory does not have enough free disk space, you can supply an alternative path . In this case 'PqDescendants' stores its intermediate results in a temporary subdirectory of . Alternatively, you can globally set the variable 'ANUPQtmpDir', for instance in your \".gaprc\"\ file, to point to a suitable location. Alternatively, you can pass 'PqDescendants' a record as a parameter, which contains as entries some (or all) of the above mentioned. Those parameters which do not occur in the record are set to their default values. Note that you cannot set both \"OrderBound\"\ and \"StepSize\". In the first example we compute all descendants of the Klein four group which have exponent-2 class at most 5 and order at most $2^6$. | gap> f2 := FreeGroup( 2, "g" );; gap> g := AgGroupFpGroup(f2 / [f2.1^2, f2.2^2, Comm(f2.2,f2.1)]); Group( g.1, g.2 ) gap> g.name := "g";; gap> l := PqDescendants( g, "OrderBound", 6, "ClassBound", 5, > "AllDescendants" );; gap> Length(l); 83 gap> Number( l, x -> x.isCapable ); 47 gap> List( l, x -> Size(x) ); [ 8, 8, 8, 16, 16, 16, 32, 16, 16, 16, 16, 16, 32, 32, 64, 64, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64 ] gap> List( l, x -> Length( PCentralSeries( x, 2 ) ) - 1 ); [ 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5 ] | In the second example we compute all capable descendants of order 27 of the elementary abelian group of order 9. Here, we supply automorphisms which form a PAG-generating sequence (in reverse order) for the class 1 group, since this makes the computation more efficient. | gap> f2 := FreeGroup( 2, "g" );; gap> g := AgGroupFpGroup(f2 / [ f2.1^3, f2.2^3, Comm(f2.1,f2.2) ]); Group( g.1, g.2 ) gap> g.name := "g";; gap> g.automorphisms := [];; gap> GroupHomomorphismByImages(g, g, [g.1, g.2], [g.1^2, g.2^2]);; gap> Add( g.automorphisms, last ); gap> GroupHomomorphismByImages(g, g, [g.1, g.2], [g.2^2, g.1]);; gap> Add( g.automorphisms, last ); gap> GroupHomomorphismByImages(g,g,[g.1,g.2],[g.1*g.2^2,g.1^2*g.2^2]);; gap> Add( g.automorphisms, last ); gap> GroupHomomorphismByImages(g, g, [g.1,g.2], [g.1,g.1^2*g.2]);; gap> Add( g.automorphisms, last ); gap> GroupHomomorphismByImages(g, g, [g.1, g.2], [g.1^2, g.2]);; gap> Add( g.automorphisms, last ); gap> l := PqDescendants( g, "OrderBound", 3, > "ClassBound", 2, > "AgAutomorphisms" );; gap> Length(l); 2 gap> List( l, x -> Size(x) ); [ 27, 27 ] gap> List( l, x -> Length( PCentralSeries( x, 3 ) ) - 1 ); [ 2, 2 ] | In the third example, we compute all capable descendants of the elementary abelian group of order $5^2$ which have exponent-$5$ class at most $3$, exponent $5$, and are metabelian. | gap> f2 := FreeGroup( 2, "g" );; gap> g := AgGroupFpGroup( f2 / [f2.1^5, f2.2^5, Comm(f2.2,f2.1)] ); Group( g.1, g.2 ) gap> g.name := "g";; gap> l := PqDescendants(g,"Metabelian","ClassBound",3,"Exponent",5);; gap> List( l, x -> Length( PCentralSeries( x, 5 ) ) - 1 ); [ 2, 3, 3 ] gap> List( l, x -> Length( DerivedSeries( x ) ) ); [ 3, 3, 3 ] gap> List( l, x -> Maximum( List( Elements(x), y -> Order(x,y) ) ) ); [ 5, 5, 5 ] | This function requires the package \"anupq\"\ (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PqList} 'PqList( )' \\ 'PqList( , )' \\ 'PqList( , )' The function 'PqList' reads a file and returns the list $L$ of ag groups defined in this file. If list is supplied as a parameter, the function returns 'Sublist( $L$, )'. If an integer is supplied, 'PqList' returns $L[]$. This function and 'SavePqList' (see "SavePqList") can be used to save and restore a list of descendants (see "PqDescendants"). This function requires the package \"anupq\"\ (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{SavePqList} 'SavePqList( , )' The function 'SavePqList' writes a list of descendants to a file . This function and 'PqList' (see "PqList") can be used to save and restore results of 'PqDescendants' (see "PqDescendants"). This function requires the package \"anupq\"\ (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{StandardPresentation} \index{automorphisms!of p groups} 'StandardPresentation( ,

, ... )' \\ 'StandardPresentation( , , ... )' Let be a finitely presented group. Then 'StandardPresentation' returns the standard presentation for the desired $p$-quotient of as an ag group. Let be the $p$-quotient whose standard presentation is computed. A generating set for a supplement to the inner automorphism group of is also returned, stored as the component '.automorphisms'. Each generator is described by its action on each of the generators of the standard presentation of . A finitely-presented group must be supplied as input. Usually, the user will also supply a prime

and the program will compute the standard presentation for the desired $p$-quotient of . Alternatively, a user may supply an ag group which is the class 1 $p$-quotient of . If this is so, a list of automorphisms of must be bound to the record component '.automorphisms' such that '.automorphisms' together with the inner automorphisms of generate the automorphism group of . The presentation for can be constructed by an initial call to Pq (see "Pq"). Of course, need not be the class 1 $p$-quotient of . However, '.automorphisms' must contain a description of the automorphism group of and this is most readily available when is an elementary abelian group. Where the necessary information is available for a $p$-quotient of higher class, one can apply the standard presentation algorithm from that class onwards. The following parameters or parameter pairs are supported. \"ClassBound\", : \\ The standard presentation is computed for the largest $p$-quotient of having lower exponent-$p$ class at most . \"Exponent\", : \\ The $p$-quotient computed has exponent . By default, no exponent law is enforced. \"Metabelian\": \\ The $p$-quotient constructed is metabelian. \"AgAutomorphisms\": \\ The automorphisms stored in '.automorphisms' are a PAG-generating sequence for the automorphism group of supplied in *reverse* order. \"Verbose\": \\ The runtime-information generated by the ANU pq is displayed. By default, pq works silently. \"OutputLevel\", : \\ The runtime-information generated by the ANU pq is displayed at output level , which must be a integer from 0 to 3. This parameter implies \"Verbose\". \"SetupFile\", : \\ Do not run the ANU pq, just construct the input file and store it in the file . In this case 'true' is returned. \"TmpDir\",

: \\ 'StandardPresentation' stores intermediate results in temporary files; the location of these files is determined by the value selected by 'TmpName'. If your default temporary directory does not have enough free disk space, you can supply an alternative path . In this case 'StandardPresentation' stores its intermediate results in a temporary subdirectory of . Alternatively, you can globally set the variable 'ANUPQtmpDir', for instance in your \".gaprc\"\ file, to point to a suitable location. Alternatively, you can pass 'StandardPresentation' a record as a parameter, which contains as entries some (or all) of the above mentioned. Those parameters which do not occur in the record are set to their default values. We illustrate the method with the following examples. | gap> f2 := FreeGroup( "a", "b" );; gap> g := f2 / [f2.1^25, Comm(Comm(f2.2,f2.1), f2.1), f2.2^5]; Group( a, b ) gap> StandardPresentation( g, 5, "ClassBound", 10 ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8, G.9, G.10, G.11, G.12, G.13, G.14, G.15, G.16, G.17, G.18, G.19, G.20, G.21, G.22, G.23, G.24, G.25, G.26 ) gap> f2 := FreeGroup( "a", "b" );; gap> g := f2 / [ f2.1^625, > Comm(Comm(Comm(Comm(f2.2,f2.1),f2.1),f2.1),f2.1)/Comm(f2.2,f2.1)^5, > Comm(Comm(f2.2,f2.1),f2.2), f2.2^625 ];; gap> StandardPresentation( g, 5, "ClassBound", 15, "Metabelian" ); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8, G.9, G.10, G.11, G.12, G.13, G.14, G.15, G.16, G.17, G.18, G.19, G.20 ) gap> f4 := FreeGroup( "a", "b", "c", "d" );; gap> g4 := f4 / [ f4.2^4, f4.2^2 / Comm(Comm (f4.2, f4.1), f4.1), > f4.4^16, f4.1^16 / (f4.3 * f4.4), > f4.2^8 / (f4.4 * f4.3^4) ]; Group( a, b, c, d ) gap> g := Pq( g4, "Prime", 2, "ClassBound", 1 ); Group( G.1, G.2 ) gap> g.automorphisms := [];; gap> GroupHomomorphismByImages(g,g,[g.1,g.2],[g.2,g.1*g.2]);; gap> Add( g.automorphisms, last ); gap> GroupHomomorphismByImages(g,g,[g.1,g.2],[g.2,g.1]);; gap> Add( g.automorphisms, last ); gap> StandardPresentation(g4,g,"ClassBound",14,"AgAutomorphisms"); Group( G.1, G.2, G.3, G.4, G.5, G.6, G.7, G.8, G.9, G.10, G.11, G.12, G.13, G.14, G.15, G.16, G.17, G.18, G.19, G.20, G.21, G.22, G.23, G.24, G.25, G.26, G.27, G.28, G.29, G.30, G.31, G.32, G.33, G.34, G.35, G.36, G.37, G.38, G.39, G.40, G.41, G.42, G.43, G.44, G.45, G.46, G.47, G.48, G.49, G.50, G.51, G.52, G.53 ) | This function requires the package \"anupq\"\ (see "RequirePackage"). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsIsomorphicPGroup} 'IsIsomorphicPGroup( , )' The functions returns true if is isomorphic to . Both groups must be ag groups of prime power order. | gap> p1 := Group( (1,2,3,4), (1,3) ); Group( (1,2,3,4), (1,3) ) gap> p2 := SolvableGroup( 8, 5 ); Q8 gap> p3 := SolvableGroup( 8, 4 ); D8 gap> IsIsomorphicPGroup( AgGroup(p1), p2 ); false gap> IsIsomorphicPGroup( AgGroup(p1), p3 ); true | The function computes and compares the standard presentations for and (see "StandardPresentation").