%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %A sisyphos.tex GAP documentation Martin Wursthorn %% %A @(#)$Id: sisyphos.tex,v 3.4 1994/06/10 02:49:26 vfelsch Rel $ %% %Y Copyright (C) 1993, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany %% %H $Log: sisyphos.tex,v $ %H Revision 3.4 1994/06/10 02:49:26 vfelsch %H updated examples %H %H Revision 3.3 1994/05/13 15:56:31 sam %H added description of 'NormalizedUnitsGroupRing' %H %H Revision 3.2 1993/11/10 10:00:14 sam %H added description of a function, minor changes %H %H Revision 3.1 1993/10/27 10:44:59 martin %H initial revision under RCS %H %% \def\SISYPHOS{\sc Sisyphos} \Chapter{Sisyphos} This chapter describes the {\GAP} accessible functions of the {\SISYPHOS} (Version~0.6) share library package for computing with modular group algebras of $p$-groups, namely a function to convert a $p$-group into {\SISYPHOS} readable format (see "PrintSisyphosInputPGroup"), several functions that compute automorphism groups of $p$-groups (see "Automorphisms"), functions that compute normalized automorphism groups as polycyclically presented groups (see "AgNormalizedAutomorphisms", "AgNormalizedOuterAutomorphisms"), functions that test two $p$-groups for isomorphism (see "IsIsomorphic") and compute isomorphisms between $p$-groups (see "Isomorphisms"), and a function to compute the element list of an automorphism group that is given by generators (see "AutomorphismGroupElements"). The {\SISYPHOS} functions for group rings are not yet available, with the only exception of a function that computed the group of normalized units (see "NormalizedUnitsGroupRing"). The algorithms require presentations that are compatible with a characteristic series of the group with elementary abelian factors, e.g.\ the $p$-central series. If necessary such a presentation is computed secretly using the $p$-central series, the computations are done using this presentation, and then the results are carried back to the original presentation. The check of compatibility is done by the function 'IsCompatiblePCentralSeries' (see "IsCompatiblePCentralSeries"). The component 'isCompatiblePCentralSeries' of the group will be either 'true' or 'false' then. If you know in advance that your group is compatible with a series of the kind required, e.g.\ the Jennings-series, you can avoid the check by setting this flag to 'true' by hand. Before using any of the functions described in this chapter you must load the package by calling the statement | gap> RequirePackage( "sisyphos" ); | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PrintSISYPHOSWord} 'PrintSISYPHOSWord(

, )' For a polycyclically presented group

and an element of

, 'PrintSISYPHOSWord(

, )' prints a string that encodes in the input format of the {\SISYPHOS} system. The string '\"1\"' means the identity element, the other elements are products of powers of generators, the -th generator is given the name 'g'. | gap> g := SolvableGroup ( "D8" );; gap> PrintSISYPHOSWord ( g, g.2*g.1 ); Print( "\n" ); g1*g2*g3 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{PrintSisyphosInputPGroup} 'PrintSisyphosInputPGroup(

, , )' prints the presentation of the finite $p$-group

in a format readable by the {\SISYPHOS} system.

must be a polycyclically or freely presented group. In {\SISYPHOS}, the group will be named . If

is polycyclically presented the -th generator gets the name 'g'. In the case of a free presentation the names of the generators are not changed; note that {\SISYPHOS} accepts only generators names beginning with a letter followed by a sequence of letters, digits,underscores and dots. must be either '\"pcgroup\"' or the prime dividing the order of

. In the former case the {\SISYPHOS} object has type 'pcgroup',

must be polycyclically presented for that. In the latter case a {\SISYPHOS} object of type 'group' is created. For avoiding computations in freely presented groups, is *neither* checked that the presentation describes a $p$-group, *nor* that the given prime really divides the group order. See the {\SISYPHOS} manual~\cite{Wur93} for details. | gap> g:= SolvableGroup( "D8" );; gap> PrintSisyphosInputPGroup( g, "d8", "pcgroup" ); d8 = pcgroup(2, gens( g1, g2, g3), rels( g1^2 = 1, g2^2 = 1, g3^2 = 1, [g2,g1] = g3)); gap> q8 := FreeGroup ( 2 );; gap> q8.relators := [q8.1^4,q8.2^2/q8.1^2,Comm(q8.2,q8.1)/q8.1^2];; gap> PrintSisyphosInputPGroup ( q8, "q8", 2 ); &I PQuotient: class 1 : 2 &I PQuotient: Runtime : 0 q8 = group (minimal, 2, gens( f.1, f.2), rels( f.1^4, f.2^2*f.1^-2, f.2^-1*f.1^-1*f.2*f.1^-1)); | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsCompatiblePCentralSeries} 'IsCompatiblePCentralSeries( )' If the component '.isCompatiblePCentralSeries' of the polycyclically presented $p$-group '' is bound, its value is returned, otherwise the exponent-$p$-central series of '' is computed and compared to the given presentation. If the generators of each term of this series form a subset of the generators of '' the component '.isCompatiblePCentralSeries' is set to 'true', otherwise to 'false'. This value is then returned by the function. | gap> g:= SolvableGroup( "D8" );; gap> IsCompatiblePCentralSeries ( g ); true gap> a := AbstractGenerators ( "a", 5 );; gap> h := AgGroupFpGroup ( rec ( > generators := a, > relators := > [a[1]^2/(a[3]*a[5]),a[2]^2/a[3],a[3]^2/(a[4]*a[5]),a[4]^2,a[5]^2]));; gap> h.name := "H";; gap> IsCompatiblePCentralSeries ( h ); false gap> PCentralSeries ( h, 2 ); [ H, Subgroup( H, [ a3, a4, a5 ] ), Subgroup( H, [ a4*a5 ] ), Subgroup( H, [ ] ) ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Automorphisms}\index{Automorphisms of $p$-groups}% \index{OuterAutomorphisms}\index{NormalizedAutomorphisms}% \index{NormalizedOuterAutomorphisms} 'Automorphisms(

)'\\ 'OuterAutomorphisms(

)'\\ 'NormalizedAutomorphisms(

)'\\ 'NormalizedOuterAutomorphisms(

)' all return a record with components 'sizeOutG':\\ the size of the group of outer automorphisms of

, 'sizeInnG':\\ the size of the group of inner automorphisms of

, 'sizeAutG':\\ the size of the full automorphism group of

, 'generators':\\ a list of group automorphisms that generate the group of all, outer, normalized or normalized outer automorphisms of the polycyclically presented $p$-group

, respectively. In the case of outer or normalized outer automorphisms, this list consists of preimages in $Aut($

$)$ of a generating set for $Aut($

$)/Inn($

$)$ or $Aut_n($

$)/Inn($

$)$, respectively. | gap> g:= SolvableGroup( "Q8" );; gap> Automorphisms( g ); rec( sizeAutG := 24, sizeInnG := 4, sizeOutG := 6, generators := [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a, b*c, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*c, b, c ] ) ] ) gap> OuterAutomorphisms( g ); rec( sizeAutG := 24, sizeInnG := 4, sizeOutG := 6, generators := [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ) ] ) | *Note*\:\ If the component '

.isCompatiblePCentralSeries' is not bound it is computed using 'IsCompatiblePCentralSeries'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AgNormalizedAutomorphisms} 'AgNormalizedAutomorphisms(

)' returns a polycyclically presented group isomorphic to the group of all normalized automorphisms of the polycyclically presented $p$-group

. | gap> g:= SolvableGroup( "D8" );; gap> aut:= AgNormalizedAutomorphisms( g ); Group( g0, g1 ) gap> Size( aut ); 4 | *Note*\:\ If the component '

.isCompatiblePCentralSeries' is not bound it is computed using 'IsCompatiblePCentralSeries'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AgNormalizedOuterAutomorphisms} 'AgNormalizedOuterAutomorphisms(

)' returns a polycyclically presented group isomorphic to the group of normalized outer automorphisms of the polycyclically presented $p$-group

. | gap> g:= SolvableGroup( "D8" );; gap> aut:= AgNormalizedOuterAutomorphisms( g ); Group( IdAgWord ) | *Note*\:\ If the component '

.isCompatiblePCentralSeries' is not bound it is computed using 'IsCompatiblePCentralSeries'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{IsIsomorphic} 'IsIsomorphic( , )' returns 'true' if the polycyclically or freely presented $p$-group and the polycyclically presented $p$-group are isomorphic, 'false' otherwise. | gap> g:= SolvableGroup( "D8" );; gap> nonab:= AllTwoGroups( Size, 8, IsAbelian, false ); [ Group( a1, a2, a3 ), Group( a1, a2, a3 ) ] gap> List( nonab, x -> IsIsomorphic( g, x ) ); [ true, false ] | (The function 'Isomorphisms' returns isomorphisms in case the groups are isomorphic.) *Note*\:\ If the component '.isCompatiblePCentralSeries' is not bound it is computed using 'IsCompatiblePCentralSeries'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Isomorphisms} 'Isomorphisms( , )' If the polycyclically or freely presented $p$-groups and the polycyclically presented $p$-group are not isomorphic, 'Isomorphisms' returns 'false'. Otherwise a record is returned that encodes the isomorphisms from to ; its components are 'epimorphism':\\ a list of images of '.generators' that defines an isomorphism from to , 'generators':\\ a list of image lists which encode automorphisms that together with the inner automorphisms generate the full automorphism group of 'sizeOutG':\\ size of the group of outer automorphisms of , 'sizeInnG':\\ size of the group of inner automorphisms of , 'sizeOutG':\\ size of the full automorphism group of . | gap> g:= SolvableGroup( "Q8" );; gap> nonab:= AllTwoGroups( Size, 8, IsAbelian, false ); [ Group( a1, a2, a3 ), Group( a1, a2, a3 ) ] gap> nonab[2].name:= "im";; gap> Isomorphisms( g, nonab[2] ); rec( sizeAutG := 24, sizeInnG := 4, sizeOutG := 6, epimorphism := [ a1, a2, a3 ], generators := [ GroupHomomorphismByImages( im, im, [ a1, a2, a3 ], [ a2, a1, a3 ] ), GroupHomomorphismByImages( im, im, [ a1, a2, a3 ], [ a1*a2, a2, a3 ] ) ] ) | (The function 'IsIsomorphic' tests for isomorphism of $p$-groups.) *Note*\:\ If the component '.isCompatiblePCentralSeries' is not bound it is computed using 'IsCompatiblePCentralSeries'. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{CorrespondingAutomorphism} 'CorrespondingAutomorphism( , )' If is a polycyclically presented group of automorphisms of a group $P$ as returned by 'AgNormalizedAutomorphisms' (see "AgNormalizedAutomorphisms") or 'AgNormalizedOuterAutomorphisms' (see "AgNormalizedOuterAutomorphisms"), and is an element of then the automorphism of $P$ corresponding to is returned. | gap> g:= TwoGroup( 64, 173 );; gap> g.name := "G173";; gap> autg := AgNormalizedAutomorphisms ( g ); Group( g0, g1, g2, g3, g4, g5, g6, g7, g8 ) gap> CorrespondingAutomorphism ( autg, autg.2*autg.1^2 ); GroupHomomorphismByImages( G173, G173, [ a1, a2, a3, a4, a5, a6 ], [ a1, a2*a4, a3*a6, a4*a6, a5, a6 ] ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{AutomorphismGroupElements} 'AutomorphismGroupElements( )' must be an automorphism record as returned by one of the automorphism routines or a list consisting of automorphisms of a $p$-group $P$. In the first case a list of all elements of $Aut(P)$ or $Aut_n(P)$ is returned, if has been created by 'Automorphisms' or 'NormalizedAutomorphisms' (see "Automorphisms"), respectively, or a list of coset representatives of $Aut(P)$ or $Aut_n(P)$ modulo $Inn(P)$, if has been created by 'OuterAutomorphisms' or 'NormalizedOuterAutomorphisms' (see "Automorphisms"), respectively. In the second case the list of all elements of the subgroup of $Aut(P)$ generated by is returned. | gap> g:= SolvableGroup( "Q8" );; gap> outg:= OuterAutomorphisms( g );; gap> AutomorphismGroupElements( outg ); [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b*c, a, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ b, a*b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a, a*b*c, c ] ) ] gap> l:= [ outg.generators[2] ]; [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ) ] gap> AutomorphismGroupElements( l ); [ GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*c, b, c ] ), GroupHomomorphismByImages( Q8, Q8, [ a, b, c ], [ a*b*c, b, c ] ) ] | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{NormalizedUnitsGroupRing} 'NormalizedUnitsGroupRing(

)' \\ 'NormalizedUnitsGroupRing(

, )' When called with a polycyclicly presented $p$-group

, the group of normalized units of the group ring $FP$ of

over the field $F$ with $p$ elements is returned. If a second argument is given, the group of normalized units of $FP / I^n$ is returned, where $I$ denotes the augmentation ideal of $FP$. The returned group is represented as polycyclicly presented group. | gap> g:= SolvableGroup( "D8" );; gap> NormalizedUnitsGroupRing( g, 1 ); &D use multiplication table Group( IdAgWord ) gap> NormalizedUnitsGroupRing( g, 2 ); &D use multiplication table Group( g1, g2 ) gap> NormalizedUnitsGroupRing( g, 3 ); &D use multiplication table Group( g1, g2, g3, g4 ) gap> NormalizedUnitsGroupRing( g ); &D use multiplication table Group( g1, g2, g3, g4, g5, g6, g7 ) | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E Emacs . . . . . . . . . . . . . . . . . . . . . local Emacs variables %% %% Local Variables: %% mode: outline %% outline-regexp: "\\\\Chapter\\|\\\\Section" %% fill-column: 73 %% eval: (hide-body) %% End: %%