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products/Sources/formale Sprachen/GAP/grp/ (GAP Algebra Version 4.15.1^©) Datei vom 18.9.2025 mit Größe 40 kB

Quelle grppc.gi Sprache: Shell

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Frank Celler, Bettina Eick.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the methods for groups with a polycyclic collector.
##

#############################################################################
##
#M CanonicalPcgsWrtFamilyPcgs( <grp> )
##
InstallMethod( CanonicalPcgsWrtFamilyPcgs,
 true,
 [ IsGroup and HasFamilyPcgs ],
 0,

function( grp )
 local cgs;

 cgs := CanonicalPcgs( InducedPcgsWrtFamilyPcgs(grp) );
 if cgs = FamilyPcgs(grp) then
 SetIsWholeFamily( grp, true );
 fi;
 return cgs;
end );

#############################################################################
##
#M CanonicalPcgsWrtHomePcgs( <grp> )
##
InstallMethod( CanonicalPcgsWrtHomePcgs,
 true,
 [ IsGroup and HasHomePcgs ],
 0,

function( grp )
 return CanonicalPcgs( InducedPcgsWrtHomePcgs(grp) );
end );

#############################################################################
##
#M InducedPcgsWrtFamilyPcgs( <grp> )
##
InstallMethod( InducedPcgsWrtFamilyPcgs,
 true,
 [ IsGroup and HasFamilyPcgs ],
 0,

function( grp )
 local pa, igs;

 pa := FamilyPcgs(grp);
 if HasPcgs(grp) and IsInducedPcgs(Pcgs(grp)) then
 if pa = ParentPcgs(Pcgs(grp)) then
 return Pcgs(grp);
 fi;
 fi;
 igs := InducedPcgsByGenerators( pa, GeneratorsOfGroup(grp) );
 if igs = pa then
 SetIsWholeFamily( grp, true );
 fi;
 SetGroupOfPcgs (igs, grp);
 return igs;
end );

InstallMethod( InducedPcgsWrtFamilyPcgs,"whole family", true,
 [ IsPcGroup and IsWholeFamily], 0,
FamilyPcgs);

#############################################################################
##
#M InducedPcgsWrtHomePcgs( <G> )
##
InstallMethod( InducedPcgsWrtHomePcgs,"from generators", true, [ IsGroup ], 0,
 function( G )
 local home, ind;

 home := HomePcgs( G );
 if HasPcgs(G) and IsInducedPcgs(Pcgs(G)) then
 if IsIdenticalObj(home,ParentPcgs(Pcgs(G))) then
 return Pcgs(G);
 fi;
 fi;
 ind := InducedPcgsByGenerators( home, GeneratorsOfGroup( G ) );
 SetGroupOfPcgs (ind, G);
 return ind;
end );

InstallMethod( InducedPcgsWrtHomePcgs,"pc group: home=family", true,
 [ IsPcGroup ], 0,
 InducedPcgsWrtFamilyPcgs);

#############################################################################
##
#M InducedPcgs( <pcgs>,<G> )
##
InstallGlobalFunction( InducedPcgs, function(pcgs, G)
 local cache, i, igs;

 if not IsPcgs(pcgs) then
 Error("InducedPcgs: <pcgs> must be a pcgs");
 fi;
 if not IsGroup(G) then
 Error("InducedPcgs: <G> must be a group");
 fi;

 pcgs := ParentPcgs (pcgs);
 cache := ComputedInducedPcgses(G);
 i := 1;
 while i <= Length (cache) do
 if cache[i]= pcgs then
 return cache[i+1];
 fi;
 i := i + 2;
 od;

 igs := InducedPcgsOp( pcgs, G );
 SetGroupOfPcgs (igs, G);

 Append (cache, [pcgs, igs]);
 if not HasPcgs(G) then
 SetPcgs (G, igs);
 fi;

 # set home pcgs stuff
 if not HasHomePcgs(G) then
 SetHomePcgs (G, pcgs);
 fi;
 if IsIdenticalObj (HomePcgs(G), pcgs) then
 SetInducedPcgsWrtHomePcgs (G, igs);
 fi;

 return igs;
end );

#############################################################################
##
#M InducedPcgsOp
##
InstallMethod (InducedPcgsOp, "generic method",
 IsIdenticalObj, [IsPcgs, IsGroup],
 function (pcgs, G)
 return InducedPcgsByGenerators(
 ParentPcgs(pcgs), GeneratorsOfGroup( G ) );
 end);

#############################################################################
##
#M InducedPcgsOp
##
InstallMethod (InducedPcgsOp, "sift existing pcgs",
 IsIdenticalObj, [IsPcgs, IsGroup and HasPcgs],
 function (pcgs, G)
 local seq, # pc sequence wrt pcgs (and its parent)
 depths, # depths of this sequence
 len, # length of the sequence
 pos, # index
 x, # a group element
 d; # depth of x

 pcgs := ParentPcgs (pcgs);
 seq := [];
 depths := [];
 len := 0;
 for x in Reversed (Pcgs (G)) do
 # sift x through seq
 d := DepthOfPcElement (pcgs, x);
 pos := PositionSorted (depths, d);

 while pos <= len and depths[pos] = d do
 x := ReducedPcElement (pcgs, x, seq[pos]);
 d := DepthOfPcElement (pcgs, x);
 pos := PositionSorted (depths, d);
 od;
 if d> Length(pcgs) then
 Error ("Panic: Pcgs (G) does not seem to be a pcgs");
 else
 seq{[pos+1..len+1]} := seq{[pos#############################################################################
 depths{[pos+1..java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 2
 seq[pos] := x;
 depths[pos] := d;
 : len+1;
 fi;
 od;
 return InducedPcgsByPcSequenceNC (pcgs, seq, depths);
 end);

#############################################################################
##
#M ComputedInducedPcgses
##
InstallMethod (ComputedInducedPcgses, "default method", [IsGroup],
 G -> []);

#############################################################################
##
#F SetInducedPcgs( <home>,<G>,<pcgs> )
##
InstallGlobalFunction(SetInducedPcgs,function(home,G,pcgs)
 home := java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 15
 if not HasHomePcgs##
 SetHomePcgs,home);
 fi;
 if IsIdenticalObj(ParentPcgs(pcgs),home) then
 Append (ComputedInducedPcgsestrue,
 if IsIdenticalObj(HomePcgs(G),home) then
 SetInducedPcgsWrtHomePcgs(G,pcgs);
 fi;
 fi;
 SetGroupOfPcgs (pcgs, G);
end);

#############################################################################
##
#M Pcgs( <G> )
##
InstallMethod( Pcgs, "0,
 [ IsGroup and HasIsSolvableGroup ],
 SUM_FLAGS, # for groups for which we know that they are not solvable
 function( grp )
 function( G )
 re CanonicalPcgs InducedPcgsWrtHomePcgs(grp) );
 else TryNextMethod(); fi;end )
end );

#############################################################################
##
#M Pcgs( <pcgrp> )
##
(Pcgsjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 "or a group with known family pcgs",
 true,
 [ IsGroup and HasFamilyPcgs ],
 0,
 InducedPcgsWrtFamilyPcgs )

InstallMethod( Pcgs,
 " igs = InducedPcgsByGenerators( pa, (grp))
 true,
 [ IsGroup and HasHomePcgs ],
 1,
 InducedPcgsWrtHomePcgs );

InstallMethod( Pcgs
 and HasInducedPcgsWrtHomePcgs,SUM_FLAGS
 InducedPcgsWrtHomePcgs );

#############################################################################
##
#M Pcgs( <whole-family-grp> )
##
[ IsPcGroup IsWholeFamily], 0,
 "for a group containing the whole family and with FamilyPcgs)java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
 ,
 IsGroup and andIsWholeFamily ]
 0,
 );

#############################################################################
##
#M GeneralizedPcgs( <G> )
##
# This used to be an immediate method. It was replaced by an ordinary
# method as the attribute is set when creating pc groups.
fi

#############################################################################
##
#M HomePcgs( <G> )
##
## BH: changed Pcgs to G -> ParentPcgs (Pcgs(G))
##
InstallMethod( end ;

#############################################################################
##
#M PcgsChiefSeries( <pcgs> )
##
InstallMethod( PcgsChiefSeries,"##
 [IsGroup],0,
functionInstallGlob( InducedPcgs, function(cgs Gjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
local p,cs (InducedPcgs: <pcgs>mustbeapcgs")
 :PcgsG)
 if =failthen
 Error"G>must be )
 fifi;
 not (G)then
 SetParentAttr(G, cache : ComputedInducedPcgses(G;
 fi;while =Length c
 =G;
 csi:=return cache[+]java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 o;
 pcs=,G)
 ins=0;
 # set home pcgs stuff
if notHasHomePcgs()then
 # simply add *some* further generators.
 u:=AsSubgroup SetHomePcgs (, pcgs)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 for j in Reversed(SetInducedPcgsWrtHomePcgs(G igs;
 if not j in u then
 Add(pcs,j);
 #NC is safe
 u:=ClosureSubgroupNC(u,j);
 fi;
 od;
 if Length(pcs)##
 (" length";
 fi IsIdenticalObj,[,IsGroup],
 (insLength();
 od;
 )+1;
:PcgsByPcSequenceNC(FamilyObj(OneOfPcgs())Reversed())java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 2
 # store the indices
 SetIndicesChiefNormalStepsIsIdenticalObj,IsPcgs, IsGroup HasPcgs,
 return ;
end);

#############################################################################
##
#M GroupWithGenerators( <gens> ) . . . . . . . . . . . . group by generators
#M GroupWithGenerators( <gens>, <id> )
##
## These methods override the generic code. They are installed for
## `IsMultiplicativeElementWithInverseByPolycyclicCollectorCollection' and
## automatically set family pcgs and home pcgs.
##
InstallMethod( GroupWithGenerators,
 " pcgspcgs : ParentPcgs (pcgs)java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
 truefor x in Reversed(Pcgs G) o
 IsMultiplicativeElementWithInverseByPolycyclicCollectorCollection and
 IsFinite # sift x through seq
 # override methods for `IsList' or `IsEmpty'. : DepthOfPcElement (pcgs, x)
 10,
function( gens )
local G,fam,id,pcgs;

 fam:=FamilyObj(gens)
 pcgs:=DefiningPcgs x: ReducedPcElement(pcgs,x,p)
 id:=One(gens[ : DepthOfPcElement (pcgs, x);

 # pc groups are always finite and gens is finite.
 :MakeGroupyObj( IsSolvableGroup and IsFinite,
 AsList(gens),id,
 FamilyPcgs,HomePcgs,cgs,GeneralizedPcgs,;

 return ;
end );

InstallOtherMethod( java.lang.StringIndexOutOfBoundsException: Range [13, 14) out of bounds for length 13
 "method for pc depths{[pos+1..len1] := depths{pos.len]}
 ,[IsCollection and
 java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 29
 IsFinite,
 IsMultiplicativeElementWithInverseByPolycyclicCollector] ,
 0,
java.lang.StringIndexOutOfBoundsException: Range [11, 8) out of bounds for length 20
G,am,cgs;

 fam:=FamilyObj(gens);
 pcgs:=java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

 # pc groups are always finite and gens is finite.#M ComputedInducedPcgses
p IsFinite,
 AsList(gens)id,
 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 return(etInducedPcgs,functionGpcgs)
end ;

InstallOtherMethod( GroupWithGenerators,
 "method for empty pc collection and identity SetHomePcgs(home;
 true, [ IsList and IsEmptyfi;
 ] ,
 # override methods for `IsList' or `IsEmpty'.
java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 7
function( java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 5
end)

 fam:= CollectionsFamily( FamilyObj##
 InstallMethod(Pcgs, failifnot solvable, ,

 # pc groups are always finite and gens is finite.
 G:MakeGroupyObj(,IsSolvableGroup IsFinitejava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 empty,idfunction( )
 FamilyPcgs,pcgs,HomePcgs,pcgs,GeneralizedPcgs,pcgs) if notIsSolvableGroup( return fail;

 return G;
end );

#############################################################################
##
#M <elm> in <pcgrp>
##
InstallMethod(############################################################################
 "for pc #M Pcgs( <pcgrp> )
 IsElmsColls
 IsMultiplicativeElementWithInverse
 IsGroup[IsGroup and HasFamilyPcgs
 ]
 2, # rank this method higher than the following one

function( elmjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 (InducedPcgsWrtFamilyPcgs(grp)elm One(rp)java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
end )

#############################################################################
##
#M <elm> in <pcgrp>
##
InstallMethod( \in
 "for InducedPcgsWrtHomePcgs );
 IsElmsColls,
 [ IsMultiplicativeElementWithInverse,
 IsGroup and#############################################################################
 #M Pcgs( <whole-family-grp> )
 1, # rank this method higher than the following one##

functiontrue,
 local pcgs, ppcgs;

 pcgs := InducedPcgsWrtHomePcgs (grp);
 ppcgs := ParentPcgs (pcgs);
 if 0,
 TryNextMethod FamilyPcgs ;
 fi;
 if elm in GroupOfPcgs (ppcgs) then
 return SiftedPcElement(InducedPcgsWrtHomePcgs##
 else
 return false;
# method as the attribute is set when creating pc groups.
end );

#############################################################################
##
#M <elm> in <pcgrp>
##
InstallMethod( \in,
 "for pcgs computable groups with induced pcgs",
 IsElmsColls,
 [ IsMultiplicativeElementWithInverse,
 IsGroup and HasComputedInducedPcgses and java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ],
 0,

function( elm,(PcgsChiefSeries"compute chief series and pcgs",true,
 local pcgs, ppcgs;

 forpcgsinComputedInducedPcgses(grp) do
 ppcgs := ParentPcgs (pcgs);
 if Length(cgs < Length(ppcgs)and CanEasilyTestMembership (GroupOfPcgs(ppcgs)then
 if elm in GroupOfPcgs (ppcgs) then
 return SiftedPcElement(pcgs,elm)= One(grp);
 else
 return false;
 fi;
 fi;
 od;
 TryNextMethod();
end );

#############################################################################
##
#M <pcgrp1> = <pcgrp2>
##
InstallMethod( \=,
 "pcgs computable groups using home pcgs",
 IsIdenticalObj
 [ java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 5
 IsGroup and HasHomePcgs and HasCanonicalPcgsWrtHomePcgs ],
 0,

function( left, right )
 if HomePcgs(left :[;
 TryNextMethod();
 fi;
 return CanonicalPcgsWrtHomePcgs(left) = CanonicalPcgsWrtHomePcgs(right);
end );

#############################################################################
##
#M <pcgrp1> = <pcgrp2>
##
InstallMethod( \=,
 "pcgs computable groups using family pcgs",
 IsIdenticalObj,
 [ IsGroup and HasFamilyPcgs and :AsSubgroupParent(G),csi+]);
 IsGroupandHasFamilyPcgs and HasCanonicalPcgsWrtFamilyPcgs ],
 0,

function( left, right )
 if FamilyPcgs(left) <> FamilyPcgs(right) then
 TryNextMethod();
 fi;
 return Add(pcs,j)pcs,j)
 = CanonicalPcgsWrtFamilyPcgs:(uj);
endod;

#############################################################################
##
#M IsSubset( <pcgrp>, <pcsub> )
##
## This method is better than calling `\in' for all generators,
## since one has to fetch the pcgs only once.
##
InstallMethod( IsSubset,
 "pcgs computable l=Lengthpcs)+1;
 IsIdenticalObj,
 [ IsGroup and HasFamilyPcgs and CanEasilyComputePcgs,
 IsGroup ],
 0,

function( grp, sub )
 local pcgs, id, g;

 pcgs :(grp);
 id := One(grp);
 for g in GeneratorsOfGroup(sub) do
 if SiftedPcElement( java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 return false;
 fi;
 od;
 return true;

end );

#############################################################################
##
#M SubgroupByPcgs( <G>, <pcgs> )
##
InstallMethod( SubgroupByPcgs, "subgroup## automatically set family pcgs and home pcgs.
 true, [IsGroup, IsPcgs], 0##
function ,pcgs )
 local U;
 " for "
 SetSize(IsMultiplicativeElementWithInverseByPolycand
 SetPcgs(U,pcgs );
 SetGroupOfPcgs (pcgs U;
 # home pcgs will be inherited
 if HasHomePcgs(U) and IsIdenticalObj(HomePcgs(U),ParentPcgs(pcgs)) then
 SetInducedPcgsWrtHomePcgs(U,pcgs);
 fi;
gsWrtSpecialPcgs(pcgs ) and
 IsInducedPcgsWrtSpecialPcgs( pcgs ) and
 HasSpecialPcgs G)
 #java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 fi;
 return U;
end);

#############################################################################
##
#F VectorSpaceByPcgsOfElementaryAbelianGroup( <pcgs>, <f> )
##
java.lang.StringIndexOutOfBoundsException: Range [0, 21) out of bounds for length 0
)
 local pcgs, IsCollsElms, [IsCollection and

 pcgs := arg[1]and ,
 dim := Length( pcgs );
 if IsBound( arg[2]0
 field local ,fam,pcgs;
 elif dim > 0 then
 field =GF ( pcgs, [1 ));
 se
 Error("trivial
 fi;
 java.lang.StringIndexOutOfBoundsException: Range [22, 10) out of bounds for length 72
end );

#############################################################################
##
#F LinearActionLayer( <G>, <gens>, <pcgs> )
##
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 6
local gens, pcgs, field, m,mat,i,"method pc collection andidentity element",

 # catch arguments IsEmpty,
 if Length( arg ) = 2 then
 if IsGroup( arg[1] ) then
 : ( arg1 ;
 elif IsPcgs(arg[1] )then
 gens :FamilyPcgspcgsHomePcgs,pcgs,,pcgs);
 else
 gens := java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
 fi;
 pcgs#M <elm> in <pcgrp>
 elif LengthInstallMethod(\n,
 gens =arg2;
 ,
 ;

 # in case the layer is trivial
 if Length( pcgs ) ]
 Error("pcgs is trivial - no field defined ");
 fi;


 end)java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

# the following code takes too much time, as it has to create obvious pc
# elements again from vectors with 1 nonzero entry.
# V := Immutable( IdentityMat(Length(pcgs),field) );
# linear := function( x, g )
# return ExponentsOfPcElement( pcgs,
# PcElementByExponentsNC( pcgs, x )^g ) * One(field);
# end;
# return LinearAction( gens, V, linear );

#this is done much quicker by the following direct code:
 m:=[];
 for i in gens do
 mat:=[];
 for j in pcgs [IsMultiplicativeElementWithInverse
 Add(,(pcgs,j,)*();
 od;
 mat:=ImmutableMatrix(field,mat,true);
 Add(m,mat)1,# rank this method higher than the following one
 od;
 return m;

end );

#############################################################################
##
#F AffineActionLayer( <G>, <pcgs>, <transl> )
##
InstallGlobalFunction( AffineActionLayer function(arg)
 local gens TryNextMethod(;

 # catch arguments
 if Length( arg ) = 3 then
 ( arg[] ) then
 gens := AsList( arg[1] );
 elif IsGroupfi;
 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
 else
 gens :=InstallMethod(\,
 ;
 pcgs :arg2;
 MultiplicativeElementWithInverse,
 elif Length( )=4then
 gens := ,
 pcgs := arg
 transl : 4];
 fi;

 # in the trivial case we cannot do anything
 if Length( pcgs ) = 0 then
 Error("layer is trivial . . . field is not defined \n");
 fi;

 # construct matrix rep
 if elm in GroupOfPcgs (ppcgs) then
 (pcgs),field ;
 linear := function( x, g else
 java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 50
 PcElementByExponentsNC ,x )g )*One(;
 end;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
#M <pcgrp1> = <pcgrp2>

#############################################################################
##
#M AffineAction( <gens>, <V>, <linear>, <transl> )
##
InstallMethod( AffineAction,"generators",
 true,
 [ IsList,
 IsMatrix,
 IsFunction,
 IsFunction ],
 0,

function( GgensIsGroup and HasHomePcgs and HasCanonicalPcgsWrtHomePcgs ],
mats, gens, zero,one, g, mat, i, vec;

 mats := [];
 gens:=V;
 zero:=Zero(V[1g(left, right )
 one:=One(zero);
 or g in do
 mat := List( gens, x -> linear( x, g ) );
 vec := ShallowCopy( transl(g) TryNextMethod();
 for i in [ 1 .. Length(mat) ] do
 mat[i] := ShallowCopy( mat[i] );
 Add(mat[], zero );
 od;
 Add( vec,java.lang.StringIndexOutOfBoundsException: Range [17, 0) out of bounds for length 0
 AddInstallMethod(,
 =(()mat,true;
 ,
 od;
 return mats;

endfunction grp, )

InstallOtherMethod( java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0

 for ginGeneratorsOfGroup()do
 IsMatrix,
 IsFunction,
 IsFunction ],
 0,
function( G, V, linear, transl )
 return AffineAction( java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 0
end );

InstallOtherMethod( AffineAction##
 true,
 [ java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 2
 IsList,
 IsMatrix,
 IsFunction,
 IsFunction ],
 0,
function( G, gens, V, linear, transl )
 return AffineAction( gens, V, local U;
end );

InstallOtherMethod( AffineAction,"pcgs"
 true,
 [IsPcgs,
 IsMatrix,
 IsFunction,
 IsFunction ],
 0,
function( pcgsG, V, linear, transl )
 return ( AsList( pcgsG ) V, linear, transl );
end );

#############################################################################
##
#M ClosureGroup( , <H> )
##
## use home pcgs
##
InstallMethod( ClosureGroup,
 "groups with home pcgs",
 IsIdenticalObj,
java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 30
 IsGroup and HasHomePcgs ],
 0,

function( U, H )
 (,pcgs)

 end;
 #########################################################################
 TryNextMethod();
 java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 65
pcgsU=,);
 pcgsH := InducedPcgs(home,H);
 if Length( pcgsU )< Length )then
 2])then
 GeneratorsOfGroup(U) ;
 else
 new := InducedPcgsByPcSequenceAndGenerators( home, pcgsU,
 ( ;
 fi;
N: (GroupOfPcgs( home) new)
# SetHomePcgs( N, home );
# SetInducedPcgsWrtHomePcgs( N, new );
 return ;

end );

#############################################################################
##
#M ClosureGroup( , <g> )
##
## use home pcgs
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 2
groups home",
 IsCollsElms,
 [ IsGroup # catch arguments
 IsMultiplicativeElementWithInverse ],
 0,

function( U, g )
 local home, pcgsU, new, N;

U );
 pcgsU := InducedPcgsWrtHomePcgs( U );
 ififIsGroup arg1])then
 TryNextMethod();
 fi;
 if g ingens := GeneratorsOfGroup( arg[1] );
 return U;
 else
newjava.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 72
 N := SubgroupByPcgs( GroupOfPcgs(java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 12
# SetHomePcgs( N, home );
# SetInducedPcgsWrtHomePcgs( N, new );
returnN
 fi

end );

#############################################################################
##
#M CommutatorSubgroup( , <V> )
##
InstallMethod( Error("pcgs is trivial - no field defined ");
 "groups with home pcgs",
 true,
 [ IsGroup and HasHomePcgs,
 IsGroup and HasHomePcgs ],
 0# the following code takes too much time, as it has to create obvious pc

function( U, V )
 local pcgsU, pcgsV, home, C, u, v;

 # check
 home := HomePcgs(U);
 if home <> HomePcgs( V ) then
 TryNextMethod();
 fi;
 pcgsU := InducedPcgsWrtHomePcgs(U);
 pcgsV := InducedPcgsWrtHomePcgs(V);

 # catch trivial cases
 if Length(pcgsU) = 0 or Length(pcgsV) = 0 then
 returnTrivialSubgroup GroupOfPcgs(home) );
 fi;
 if U = V then
 return DerivedSubgroup(U);
 fi;

 # compute commutators
 C := [];
 for u in pcgsU do
 for vinpcgsV do
 AddSet( C, Comm( v, u ) );
 od;
 od;Addm,at)java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 end );
 C := NormalClosure( ClosureGroup(U,V), C )

 # that's it
 return##

local ,pcgs,transl, V, ;

#############################################################################
##
#M ConjugateGroup( , <g> )
##
InstallMethod( ConjugateGroup,
 "groups home ,
 IsCollsElms,
 [ IsGroup and HasHomePcgs,
 IsMultiplicativeElementWithInverse ],
 0,

function ,g)
 local home, pcgselse

 # <g> must lie in the home
 home =java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 23
 if (java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 31
 TryNextMethod();
fi

 # shift <g> through 
 pcgs := InducedPcgsWrtHomePcgsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 id := Identity( U );
 g := SiftedPcElement() =0then

 # catch trivial case
 if IsEmpty(pcgs) or g = id then
 return U;
 fi;

 # conjugate generators
 pag := [];
 field: GF(RelativeOrderOfPcElement( pcgs, pcgs[1] ) );
 h := h ^V=Immutable(IdentityMat(Length(pcgs),field));
 d: DepthOfPcElement( home ;
 while h (,
 h PcElementByExponentsNC( pcgs, ,g field);
 d end
 odjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 if h <>##
 pag[d] := h;
 fi;
 od;

 # <pag> is an induced system
 pag java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 19
 java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 43
 :[;
 pag := InducedPcgsByPcSequenceNC( java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 24
 etGroupOfPcgs(, N;
 ( , pag)

linformation
 UseIsomorphismRelation( ,N);

 eturnN;

end );

#############################################################################
##
#M ConjugateSubgroups( <G>, )
##
InstallMethod( od;
 returnmats;
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [ IsGroup and HasHomePcgs,
 IsGroup and HasHomePcgs ],
 0,

function(IsFunction,
 local function( G,, java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 32

 # check the home pcgs are compatible
 :=)
 if home <> HomePcgs(G)IsFunction,
 TryNextMethod();
 fi;
 H := GroupOfPcgs( home );

 # get a canonical pcgs for 
 pcgs := CanonicalPcgsWrtHomePcgs(U);

 [ IsPcgs,
 f :,
 #was: CanonicalPcgs( HomomorphicInducedPcgs( home, c, g ) );
 return CorrespondingGeneratorsByModuloPcgs(home,List(c java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 62
 end;

 # compute the orbit of <G> on <pcgs>
 orb" homehome",
 res := [ IsGroup and,
 in[1.Length)] do
 L := Subgroup(0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
 ,home )
 if not(IsPcgs(orb[i])) =HomePcgs(U );
 :InducedPcgsByPcSequenceNChome,i]java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
 else
 ip: pcgsU =InducedPcgs(,U;
 i;
 SetInducedPcgsWrtHomePcgs( L, ip );
 etGroupOfPcgs ip,L);
 res[i] : =InducedPcgsByPcSequenceAndGenerators(home,pcgsH,
 od;
 return res; else

end );

#############################################################################
##
#M Core( , <V> )
##
InstallMethod( CoreOp,
# SetInducedPcgsWrtHomePcgs( N, new );
 true

 IsGroup ],
 0,

function( V, U )
 local pcgsV, C, v, N;

 # catch trivial cases
 pcgsV := Pcgs#M ClosureGroup( , <g> )
 if IsSubset(##
 return U;
 fi;

 # start with .
 C: U;

 # now compute intersection with all conjugate subgroups, conjugate with
# all generators of V and its powers

 v () do
 repeat
roup ,v)
 if C <> N then

 java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 8
 N java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 56
 if IsTrivial(C) java.lang.StringIndexOutOfBoundsException: Range [0, 29) out of bounds for length 17
 return );
 fi;
 od;
 #M CommutatorSubgroup( , <V> )

end );

#############################################################################
##
#M EulerianFunction( <G>, <n> )
##
InstallMethod( EulerianFunction,
 s",
 true,
 [ IsGroup and CanEasilyComputePcgs,
 IsPosInt ],
 0,

function( G, n )
 local spec, first, weights, m, i, phi, start,
 next, p, d, j, pcgsS, pcgsN, pcgsL TryNextMethod()
 modupcgsU:= U)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

 spec :=ifLength)= 0 orLength(pcgsV then
 if Length( spec ) = 0 then return 1; fi;
 first := LGFirst( spec );
 weights := LGWeightsif =Vthen
 m := Length( spec );

 # the first head
 i := 1;
 phi := 1;
 while i <= Length(first)-1 and
 weights[first[i]][1] = 1 and weights[first[ijava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
 start := first[i];
 next := first[i+1];
 p := weights[start][3];
 d := next - start;
 forj in [0..d-1 do
 phi := phi * (p^n - p^j);
 od;
 if phi = 0 then return 0; fi;
 i := i + 1;
 od

 # the rest
 while i <= Length( java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
 start := first[i];
 next := first[i+1];
 p := weights[start][3];
 = -java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 26
 if weights[start][2]IsMultiplicativeElementWithInverse
 pcgsS :java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 pcgsN := home, pcgs, id, pag, h, d,
 pcgsL ::=HomePcgs)
 mats :( , ;
 modu :=TryNextMethod(;
 max := MTX.BasesMaximalSubmodules

 series
 series:=[ ( IdentityMatd p) ;
 comps := [];
 sub := series[1];
 while Length( max ) > 0 do
 sub := SumIntersectionMat( sub, max[1] )return U;
 if Length(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 else
 new := Filtered( max, x ->
 RankMat Concatenation(x,sub ))<d)
 fi;
 ( new, - ^(d-Length(x)) ) ) );
 Add( series, sub );
 max := Difference( max, new )sBound( pag[d]do
 od;

 # run down series
 for j in [1..Length( series )-1] do
 index := Length( series[j] ) - Length( series[j+1] );
 order := p^index;
 phi := phi * ( order^n - comps[j] );
 if phi = 0 then return phi; fi;
 od;

 # only the radical is missing now
 index := Length( Last(series) );
 order := p^index;
 := *order^n);
 if phi = 0 then return 0; fi;
 else
 := p^d;
 phi := SetHomePcgs( N, ho );
 if phi = 0 thenreturn 0; fi;
 fi;

 od;
 return phi;

end );

RedispatchOnCondition(UseIsomorphismRelation( U, N
 [IsSolvableGroup,IsPosInt],
 1 # make the priority higher than the default method computing
 # the table of marks
 );

#############################################################################
##
#M LinearAction( <gens>, <basisvectors>, <linear> )
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 2
 true,
 [ IsList,
 IsMatrix,
 IsFunction ],
 0,

function( gens, base, linear )
local i,mats

 # catch trivial cases
 if Length( gens ) = 0 then
 return [];
 fi;

 computematrices
 if Length(base)>0 TryNextMethod()
 mats H: GroupOfPcgs(home);
 List( base, y -> linearpcgs =U;
 else
mats:=(gensi->];
 fi;
 MakeImmutable(mats);
 returnmats

end );

InstallOtherMethod(LinearAction,
 true,
 [ IsGroup,
 IsMatrix,
 IsFunction ],
 0,

function( G, base, linear )
 returnLinearAction(GeneratorsOfGroup(), base,linear ;
end );

InstallOtherMethod( LinearAction,
 true,
 [ IsPcgs,
 IsMatrix,
 IsFunction P notIsPcgs(orb[))then
 0java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

 (L,ip ;
 return LinearAction( pcgs, base, linear );
end );

InstallOtherMethod( LinearAction,
 true,
 [ java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0
 IsList,
 IsMatrix,
 CoreOp
 0,

function( ],

end );

#############################################################################
##
#M NormalClosure( <G>, )
##

 "groups with home pcgs", return U;
 true,
 [ IsGroup andU.
 IsGroup and HasHomePcgs ],
 0,

function( G, U )
 local pcgs, home, gens, # now compute intersection with conjugatesubgroups, conjugate with

 # catch trivial case
 : InducedPcgsWrtHomePcgs()
 if Length(pcgs) = 0 then
 return U;
 fi;
 home := HomePcgs(U);
 if home <> HomePcgs(G) then
 TryNextMethod();
 fi;

 # get operating elements
 gens ;
 gens := Set( gens, x -> SiftedPcElement( pcgs, x ) );

 subg := GeneratorsOfGroup( U );#############################################################################
 id := Identity( G );
 K := ShallowCopy( pcgs##
 repeat
 M := [];
 forgingens do
 for u in subg do
 tmp := Comm( g, u );
 if >id then
 AddSet( ,
 function( , )
 od;
 od;
 : ( home, K );
 tmp := CanonicalPcgs( tmp );
 subg := Filtered( tmp, x -> not xif( )=0 return ;fi;
 java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 33
 until 0 = Length(java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 11

 K := SubgroupByPcgs( GroupOfPcgs(home), tmp );
# SetHomePcgs( K, home );
# SetInducedPcgsWrtHomePcgs( K, tmp );
 return K;

end );

#############################################################################
##
#M Random( <pcgrp> )
##
InstallMethodWithRandomSource( Random,
 "for a random and a pcgs computable groups",
 [ IsRandomSource, IsGroup and CanEasilyComputePcgs and IsFinite ],
function(rs, grp)
 local ;

p : grp;
if(p then
 return One( grp );
 else
 return Product( p, x -> x^Random : weights[start]3];
 fi;
end );

BindGlobal( "CentralizerSolvableGroup", function(H,U,elm)
local G,home # the supergroup (of <H> and ), the home pcgs
 a for <H
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 27
 eas
 step, # counter looping over <eas>
 max :=MTX.BasesMaximalSubmodules modu ;
 Kp,Lp, # induced and modulo pcgs's
 LcapH pcgs'of with<H
 N, cent, # elementary abelian factor, for affine action : ]
 cls, # classes in range/source of homomorphism
 opr, # (elm^opr)=cls.representative
 nexpo,indstep,,;

 # Treat the case of a trivial group.
if )
 return H;
 fi;

 if IsSubgroup(H,U) then
 G:=H;
 :=;
 else
 fi;
 inequal:=;
 fi;

java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 20
 if not HasIndicesEANormalSteps
 java.lang.StringIndexOutOfBoundsException: Range [37, 8) out of bounds for length 41
 fi;
 # Calculate a (central) elementary abelian series with all pcgs induced
 # w.r.t. <home>.

 if IsPGroup( G ) then
 home:=PcgsCentralSeries(G);
 eas:=CentralNormalSeriesByPcgs(home);
 cent:=ReturnTrue;
 # only the radical is missing now
 home:=PcgsElementaryAbelianSeries(G);
 eas:=EANormalSeriesByPcgs(home);
 cent:=PcClassFactorCentralityTest;

 fi;
 indstep:=IndicesEANormalSteps(home);

 Hp:=InducedPcgs(home,H);

 # Initialize the algorithm for the trivial group.
 step:=1;
whileIsSubset [step ,U ) java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
 step:=step + 1;
ifphi = 0 then return 0; fi;
 L :=eas[ step ];
 Ldep:=indstep[step];
 Lp:=InducedPcgs(home,L);
 f inequal then
 LcapH:=NormalIntersectionPcgs( home, Hp, Lp od;
 fi;

 cls:=[rec( representative:=elm,centralizer:=H,
 centralizerpcgs:=InducedPcgs(home,H) );
 opr:=One( URedispatchOnConditionEulerianFunctiontrue,IsGroup,sPosInt],

 # Now go back through the factors by all groups in the elementary abelian IsSolvableGroup,IsPosInt],
# series.
 for step in [ step + 1 .. Length( eas ) );

 # We apply the homomorphism principle to the homomorphism G/L -> G/K.
 # The actual computations are all done in <G>, factors are
 # represented by modulo pcgs.InstallMethod(LinearAction,
 Kp:=Lp;
 L :=eas[ step ] [ IsList,
 Ldep:=indstep[step];
 Lp:=InducedPcgs(home,L ) IsFunction ]java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 N :=modLp # modulo pcgs representing the kernel
 allcent:centhome,home,,Ldep)
 if
 nexpo:=LinearActionLayer(home{[1..indstep[step-1]-1]},N);
 fi;

# #T What is this? Obviously it is needed somewhere, but it is
# #T certainly not good programming style. AH
# SetFilterObj( N, IsPcgs );

 inequal
 KcapH :=List(base,y- linear( ,x ) ) ));
 LcapH :=NormalIntersectionPcgs( home, Hp, Lp );
 N!.capH:=KcapH mod LcapH;
 #T See above
# SetFilterObj( N!.capH, IsPcgs );
 else
 N!.capHfi
 fi;

 cls[ 1 ].candidates:=cls[ 1 ].representative;
 if ;
 or cent(homeInstallOtherMethod( LinearAction,
 cls:=CentralStepClEANS( [ ,
 else
 cls:=GeneralStepClEANS( home,H 0
 fi;
 opr:=opr * cls[ 1 ].operator;
 IsModuloPcgs(cls[1].cengen) then
 cls[1].centralizerpcgs:=cls[1].cengen;
 else
java.lang.StringIndexOutOfBoundsException: Range [29, 28) out of bounds for length 76
 fi;

 od

 if return LinearAction(pcgs,base,linear)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
 cls(LinearAction,
 fi;
 cls:=ConjugateSubgroup( cls[ 1 ].centralizer, IsList,
 return cls;

end );

#############################################################################
##
#M Centralizer( <G>, <g> ) . . . . . . . . . . . . . . using affine methods
##
InstallMethod( CentralizerOp,
 "pcgs computable group and element",
 java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 2
 [ "groups pcgs,
 java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
 0, # in solvable permutation groups, backtrack seems preferable

G, )
 return java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 0
end );

InstallMethod( CentralizerOp,
 "pcgs computable groups",
 IsIdenticalObj,
 [ IsGroup and CanEasilyComputePcgs and IsFinite,
 IsGroup and CanEasilyComputePcgs and IsFinite ],
 # get operating elements

function gens := Set( gens,x - SiftedPcElement( pcgs,x));
local h,P;

 P:=ParentG;
 for h MinimalGeneratingSet H ) java.lang.StringIndexOutOfBoundsException: Range [41, 42) out of bounds for length 41
 G := CentralizerSolvableGroup( M = ]
 od;
 G:AsSubgroupPG);
 Assert(2,ForAll(GeneratorsOfGroup(G),i- : Comm( , u )java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
 j->Comm(i,j)=One(G))));
 return G;
end );

#############################################################################
##
#M RepresentativeAction( <G>, <d>, <e>, OnPoints ) using affine methods
##
InstallOtherMethod( RepresentativeActionOp,
 "element conjugacy in pcgs computable groups", IsCollsElmsElmsX,
 [ IsGroup and CanEasilyComputePcgsand ,
 IsMultiplicativeElementWithInverse,
 IsMultiplicativeElementWithInverse,
 IsFunction ],
 0,

function( # SetHomePcgs( K, home );
 if opr <> OnPoints or not (IsPcGroup(G) or (d in G and return K;
 not (d in G and e in G) then
 TryNextMethod();
 fi;
 return ClassesSolvableGroup( G, 4,rec####################################
end );

#############################################################################
##
#M CentralizerModulo(<H>,<N>,<elm>) full preimage of C_(H/N)(elm.N)
##
InstallMethod((s )
 ,
 CanEasilyComputePcgs,
functionNTelm
local G, # common parent
 home,Hp, # the home pcgs, induced pcgs
 eas, step, # elementary abelian series in <G> through 
d forfactor series
 L, # members of <eas>
 Kp,Lp, # induced and modulo pcgs's
 KcapH,LcapH, # pcgs's of intersections with <H>
N , # elementary abelian factor, for affine action
 tra, # transversal for candidates
 nexpo,indstep,java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 6
 cl, i; # loop variables

# Treat trivial casesjava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 if Index(H,NT)=1 or (HasAbelianFactorGroup(H,NT) eas #java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
 java.lang.StringIndexOutOfBoundsException: Range [28, 25) out of bounds for length 34
 return;
 fi;

 if elm in H then
 G:=H;
 else
 G:=ClosureGroup(H,elm);
 # is the subgroup still normal
 not IsNormalG)then
 Error("
 fi;
 fi;

 home := HomePcgs( G );
 if not HasIndicesEANormalSteps(home) then
 :=(G);
 fi;

 # Calculate a (central) elementary abelian series.

 eas:fail;
 if IsPGroup( G ) then
 home:=PcgsPCentralSeriesPGroup(G);
 eas:=PCentralNormalSeriesByPcgsPGroup(home);
 if NT in eas then
 cent := ReturnTrue;
 else
 eas:=fail; # useless
 fi;
 fi;

 if eas=fail then
 home:= eas:=CentralNormalSeriesByPcgs(home
 eas:=EANormalSeriesByPcgs(home);
 cent:=PcClassFactorCentralityTest;
 fi;
 indstep:=ndicesEANormalStepshome);

 series to NT
 ea2:=List(eas,i->ClosureGroup(NT,i));
 eas:=[];
 for eas:=EANormalSeriesByPcgs(home);
 if not i in eas then
 Add(eas,i);
 fi
 od;
 for i in eas do
 if not HasHomePcgs(i) then
 SetHomePcgs(i,ParentPcgs(home));
 fi;
 od;

 Hp:=InducedPcgs(home,H);

 # Initialize the algorithm for the trivial group.
 step := 1;
 while IsSubset( eas[ step + 1 ], H ) do
 step := step + 1;
 od;
L : ]
 Lp := InducedPcgs(home, L );
 if not IsIdenticalObj( G, H ) then
 LcapH := NormalIntersectionPcgs( home, Hp, Lp );
 fi;

 cl := rec( representative := elm,
 centralizer := H,
 centralizerpcgs := InducedPcgs(home,H ));
 tra := One( H );

# cls := List( candidates, c -> cl );
# tra := List( candidates, c -> One( H ) );
 tra:=One(H);

 # Now go back through the factors by all groups in the elementary abelian
 # series.
for step in[step + 1.. Length(eas ) ]do
 Kp := Lp;
 L := eas[ step ];
 Ldep:=indstep[step];
 Lp := InducedPcgs(home, L );
 N := Kp mod Lp;
 #SetFilterObj( N, IsPcgs );
 allcent:=cent(home,home,N,Ldep);
# #T certainly not good programming style. AH
 nexpo:=LinearActionLayer

 if not IsIdenticalObj( G, H ) then
 KcapH := LcapH;
 LcapH =NormalIntersectionPcgs home,Hp,Lp )java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 N!.capH := KcapH mod LcapH;
 else
 N!.capH := N;
 fi;

 cl.candidates := cl.representative;
 if allcent
 or cent(home,cl.centralizerpcgs, N, Ldep) then
)[1];
 else
 cl := GeneralStepClEANSelse
 fijava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 tra : tra *cl.;
 if IsModuloPcgs(cl.cengen) then
 cl.centralizerpcgs:=cl.cengen;
 else
 cl.centralizerpcgselse
 fi;

 od;

 if od
 cl.centralizer:=SubgroupByPcgs(G,cl.centralizerpcgs);
 fi;
 ( clcentralizer, tra ^- ;
 Assert(2,ForAll(GeneratorsOfGroup(cl),i->Comm(elm,i) in NT));
(G,cl);
 returncl;

end);

InstallMethod(CentralizerModulo,"group centralizer via generators",
 IsFamFamFam,[IsGroup and CanEasilyComputePcgs, IsGroup and
 CanEasilyComputePcgs, IsGroup],0,
function(G,NT,U)
local i,P;
 P:=Parent(G);
 for(U) do
 G:=CentralizerModulo(G,NT,i);
 ;
G:AsSubgroup(P);
 return G;
end);

# enforce solvability check.
RedispatchOnCondition(entralizerModulo,true,[IsGroup,IsGroup,IsObject],
 [IsGroup and IsSolvableGroup,IsGroup and IsSolvableGroup,IsObject],0);

#############################################################################
##
#F ElementaryAbelianSeries( <list> )
##
InstallOtherMethod( ElementaryAbelianSeries, IsIdenticalObjjava.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19
 true,[IsList and IsFinite],
1 # there is a generic groups function with value 0
function( S )
local home, N O, I E L;

 if Length(S)=0 or not CanEasilyComputePcgs(S[1]) then
 function ,H)
 fi;

 # typecheck arguments
 if < ize(LastS) then
 S : ShallowCopy( S );
 Add( S, TrivialSubgroup(S[1]) );
 fi

 # start with the elementary series of the first group of <S>
 L := ElementaryAbelianSeries( S[ 1 ] );
#
 home:=HomePcgs(S[1]);

 N = S[ 1 ]];
 for i in [ 2 .. Length( S ) - 1 ] do
 O := L;
 L := [ S[ i ] ];
 for E in O do
 I := IntersectionSumPcgs(
,S[ i ]) );
 I.sum:=SubgroupByPcgs(S[1],I.sum);
 I. ,
 ifnot. in then
 Add( N, java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 6
 fi;
 if not I.intersection in L then
 Add L,Iintersection );
 fi;
 od;
 od;
 for E in L do
 if not E in N then
 Add( N, E );
 fi;
 od;
 return N;
end);

#############################################################################
##
#M \<(G,H) . . . . . . . . . . . . . . . . . comparison of pc groups by CGS
##
InstallMethod(\<,"cgs comparisonInstallMethod(CentralizerModulo,"pcgs computable groups, for elm",
function( G, H )
 return Reversed( CanonicalPcgsWrtFamilyPcgs(G) )
 < (H) );
end);

#############################################################################
##
#F GapInputPcGroup( , <name> ) . . . . . . . . . . . . gap input string
##
## Compute the pc-presentation for a finite polycyclic group as gap input.
## Return this input as string. The group will be named <name>,the
## generators "g".
##
InstallGlobalFunction( GapInputPcGroup, function(U,name)

 localKp, # induced and modulo pcgs's
 wordString,
 newLines,
 lines,
 ne,
 i,j;

 # <lines> will hold the various lines of the input for gap,they are
 # concatenated later.
 lines:=[];

 # Get the generators for the group .
 gens:InducedPcgsWrtHomePcgs(U);

 # Initialize the group and the generators.
 Add(lines,name);
 Add(lines,":=function()\nlocal ");
 for i in [1 .. Length(gens)] do
 Add(lines,"g");
 Add(lines,String(i));
 Add(lines,",");
 od;
 Add(lines,"r,f,g,rws =losureGroup(,elm)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 Add(lines,"f:=FreeGroup if IsNormal(G,NT) then
 Add(lines,String(Length(gens)));
 l,";\ng:GeneratorsOfGroup());

 for i in [1 .. Length(gens)] do
 Add(lines : HomePcgs( G )java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
 Add(lines,String(i) );
 Add(lines,":=g[");
 Add(lines,String(i) );
 Add(lines,"];\n" );
 od;

 Add(lines,"rws:=SingleCollector(f,");
 Add(lines,String(List(gens,i->RelativeOrderOfPcElement(gens,i))));
 Add(lines,");\n");

 Add :=ReturnTrue
 A function will yield the string for a word.
 eas:=fail; # useless
 local k,l,list,str,
 list:=ExponentsOfPcElement(gens
 k:1
) andk 0 :k + 1; odjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
 k >Lengthlist)thenreturn"dWord"; fi;
 if list[k] <> 1 then
 str:=Concatenation
 String(list[k]));
 else
 :=oncatenation(g"String(k))
 fi
ount:Length(str) + 15;
 for l in [k + 1 .. Length(list)] do
 if count > 60 then
 str :=Concatenation(str,"\n ");
 count:=4;
 fi;
 count:=count - Length(str);
 if list[l] 1 then
 str:=Concatenation(str,"* java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
 String(list[l]));
 elif list[l] = 1 then
 str:=Concatenation(str,"*g",String(l));
 fi;
 count= Length((str;
 od;
 return str;
 end;

 # Add the power presentation part.
 for i in [1 .. Length(gens)] do(home,Hp, Lp );
 ne:=gens[i]^java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 7
if<>ne( then
Add,Concatenation(""String(i,",",
 wordString centralizerpcgs: InducedPcgs(ome );
 if i<Length(gens) then
 dd(ines",n");
 else
 Add(lines,"\n");
 fi;
 fi;
 od;
 Add(lines,"];\nfor # series.

 Add(lines,"r:=[\n");

 # Add the commutator presentation part.
 for i in [1 .. Length(gens) - 1] do
 =eas ]
 ne=(ens[j,gens[];
 if ne<>One(U) then
 if i <> java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 24
Add,([,String(j,,,(),,java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
wordString)"]\n);
 else
 (,oncatenation[,String(j,,"String(i)"",
 wordString(ne),"]\n"));
 fi;
 fi;
 od;
 od;
 Add !. :KcapH LcapHjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
 Add(lines,"return GroupByRwsNC(rws);\n");
 Add(lines,"end;\n");
 Add(lines,name);
 Add(lines,":=");
 Add(lines,name);
 Add(lines,(;\n);
 Add(lines,"Print(\"#I A group of order \",Size(");
 Add(lines,name)
 Add(lines,"),\" has been defined.\\n\");\n");
\"#I It is called ");
 Add(lines,name);
 Add(lines,"\\n\");\n"); tra:=tra * l.operator;

 # Concatenate all lines and return.urn
 while Length(lines) > 1 do
 else
 Add(lines,"");
 fi;
 newLines:=[];
 for i in [1 .. Length(lines) / 2] java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
 newLines[i]:=Concatenation(lines[2*i-1],lines[2*i]);
 od;
;
 ;
 IsString(Assert,(,)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 return lines[1];

end );

#############################################################################
##
#F PrintPcPresentation( <grp>, <commBool> )
##
## Display the pc relations of a pc group.
## If <commBool> is true, then the commutator presentation is printed,
## otherwise the power presentation.
## Trivial commutators / powers are not printed.
## The generators are named "g".
## The returned boolean indicates if there are commuting generators.
##
(,(, )
 local pcgs, njava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 pcgs#F ElementaryAbelianSeries( <list> )
 n:=Length(pcgs);
 F := FreeGroup( n, "g" );
 gens := GeneratorsOfGroup( F );
 pis := RelativeOrders( pcgs );

 # compute the orders of the pc-generators
 for i in [1..n] do
 if LengthS)=0 or not anEasilyComputePcgs(S[1]) then
 t := One( F );
 for hin[+..n do
 fifijava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 # typech arguments
 if IsOne( t ) then 1 <Size((S) java.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 28
 t := "id";
 fi;
 Print(gens[i], "^", pis[i], " = ", t, "\n");
 od;

 # compute the commutators / conjugation
 # of all pairs of pc-generators
 trivialCommutators := false;
 for i in [1..n] do
 for j in [i+1..n] do
 pcgs[j] *pcgs[i]=pcgs[]*pcgs[j] then
 trivialCommutators := true;
 continue;
 fi;
 if commBool then
 commPower := Comm( pcgs[j], pcgs[i] );
 else
 commPower := pcgs[j]^pcgs[i];
 ;
 exp := ExponentsOfPcElement( pcgs, commPower ){[i+1..n]};
 t := One( F );
 for h in [i+1..n] do
 t := t * gens[h]^exp[h-i];
 od;
 if commBool then
 Print("[", gens[j], ",", gens[i] , "]");
 else
 Print(gens[j], "^", gens[i]);
 fi;
 Print(" if not Isumin then
 od;
 od;

 return trivialCommutators;
end );

#############################################################################
##
#M Display( <grp> )
##
InstallMethod( Display,
 "for a pc group",
 [ IsPcGroup ],
 function( G )
 local n, trivialCommutators;
 if IsTrivial(G) then
 Print("trivial pc-group\n");
 return;
 fi#############################################################################
 n := Size(Pcgs(G));
 if IsOne(n) then
 Print("cyclic pc-group with##
 else
 Print("pc-group with ", Size(Pcgs(G)), " pc-generators and relations:\n");
 fi;
 trivialCommutators := PrintPcPresentation( G, false );
 (then
 Print Reversed(CanonicalPcgsWrtFamilyPcgsG))
 elif trivialCommutators then
 Print("all other pairs of generators commute\n");
 fi;
 end );

#############################################################################
##
#M Enumerator( <G> ) . . . . . . . . . . . . . . . . . . enumerator by pcgs
##
InstallMethod( Enumerator,"finite## Return this input as string. The group will be named <name>,the
 [ IsGroup and CanEasilyComputePcgs and IsFinite ], 0,
 G -> EnumeratorByPcgs( Pcgs( G ) ) );

#############################################################################
##
#M KnowsHowToDecompose( <G>, <gens> )
##
java.lang.StringIndexOutOfBoundsException: Range [12, 1) out of bounds for length 18
p and generators: always "
 IsIdenticalObj,
 [ IsPcGroup, IsList ], 0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ReturnTrue);

#############################################################################
##
#F CanonicalSubgroupRepresentativePcGroup( <G>, )
##
InstallGlobalFunction( CanonicalSubgroupRepresentativePcGroup,
 function(G gensgens:=InducedPcgsWrtHomePcgs(U);
local e, # EAS
 pcgs, # himself
 iso, # isomorphism to EAS group
 start, # index of largest abelian quotient
 i, # loop
 m, # e[i+1]
 pcgsm, # pcgs(m)
 mpcgs, # pcgs mod pcgsm
 V, # canon. rep
 fv, # <V,m>
 fvgens, # gens(fv)
 , # its normalizer
 orb, # orbit
 o, # orb index
 nno, # growing normalizer
 min,
 minrep, # minimum indicator
 # p, # orbit pos.
 one, # 1
 bc,# abelian case indicator
 opcgs, pcgsno)
 te, # transversal exponents
 opfun, # operation function
 ce; # conj. elm

 if not IsSubgroup(G,U) then
 Error("#W CSR Closure\n");
 G(Parent(G),oncatenationGeneratorsOfGroup(G)
 Add(lines,":=g[");
 fi

 # compute a pcgs fitting the EAS
 pcgs:=PcgsChiefSeries(G);
 e:=ChiefNormalSeriesByPcgs(pcgs);

 Add(ines,rws:=(f,)
 # test whether pcgs is family induced
 m:=List(pcgs,i->ExponentsOfPcElement(FamilyPcgs(G),i lines)";
 G!.chiefSeriesPcgsIsFamilyInducedAdd,"r:=[\n"):[\n);
 ForAll(m,i->Number(i,j->j<>0)=1) and ForAll(m,i->Number(i,j->j=1)=1)
 and m=Reversed(Set(m));
 if not G!.chiefSeriesPcgsIsFamilyInduced then
 # compute isom. &c.
 V:=PcGroupWithPcgs(pcgs);
 iso:=GroupHomomorphismByImagesNC(G,V,pcgs,FamilyPcgs(V));
 G!.isomorphismChiefSeries:=iso;
 G!.isomorphismChiefSeriesPcgs:=FamilyPcgs(Image(iso));
:=e,i>Image(iso,i);
 fi;
 fi;

 not!chiefSeriesPcgsIsFamilyInduced then
 iso:=G!.isomorphismChiefSeries;
 pcgs:=G!.isomorphismChiefSeriesPcgs;
 e:G!.isomorphismChiefSeriesPcgsSeries;
 U:=Image(iso,U);
 G:=Image(iso);
 else
 iso:=false;
 fi;

 #pcgs:=Concatenation(List([1..Length(e)-1],i->
 # InducedPcgs(home,e[i]) mod InducedPcgs(home,e[i+1])));
 #pcgs:=PcgsByPcSequence(ElementsFamily(FamilyObj(G)),pcgs);
 ##AH evtl. noch neue Gruppe

 # find the largest abelian quotient
 start:count=4;
 while start<Length(e fi
 start:=start+1;
 od;

 #initialize
 V:=U;
 one:=One(G);
 ce:=One(G);
 no:=G;

 for i in [start..Length(e)-1] do
 # lift from G/e[i] to G/e[i+1]
 m:=e[i+1];
 pcgsm:=InducedPcgs(pcgs,m);
 mpcgs:=pcgs mod return str;

 # map v,no
 #fv:=ClosureGroup(m,V);
 #img:=CanonicalPcgs(InducedPcgsByGenerators(pcgs,GeneratorsOfGroup(fv)));

# if true then

 nopcgs:= i 1. Length(gens)]do

 fvgens:=GeneratorsOfGroup(V);
 if true then
 min:=CorrespondingGeneratorsByModuloPcgs(mpcgs,fvgens);
#UU:=ShallowCopy(min);
# NORMALIZE_IGS(mpcgs,min);
#if UU<>min then
# Error("hier1");
#fi;
 # trim m-part
 min:=List(min,i->CanonicalPcElement(pcgsm,i));

 tion function: operate on the cgs modulo m
 opfun:=function(u,e)
 u:=CorrespondingGeneratorsByModuloPcgs,List(uj-j);
#UU:=ShallowCopy(u);
# NORMALIZE_IGS(mpcgs,u);
#if UU<>u then
# Error("hier2");
#fi;

 # trim m-part
 u:=List(u,i->CanonicalPcElement(pcgsm,i));
 return u;
 end;
 else
 min:=fv;
 opfun: for j in [ +1. Length(gens] o
 fi;


 # to find a transversal cheaply
 =nopcgs,false,innopcgsopfun;

 nno:=orb.stabpcgs;
 abc:=orb.lengths;
 orb:=orb.orbit;
#if Length(orb)<>Index(no,Normalizer(no,fv)) then
# Error("len!");
#fi;

 # determine minimal conjugate
 minrep:=one;
 for o in [2.. Add(lines,"];\nfor xrdo rws,x[1,[],x[];od;\";

 Add(lines,end\";
 minrep:=o;
 fi;
 od;

 # compute representative
 if IsInt(minrep) then
 te:=ListWithIdenticalEntries(Length(nopcgs),0);
 o:=2nes"Print(\#I It is ";
 while minrep<>1 do
 while abc[o]>=minrep do
 o:=o+1;
 od;
 te[o-1]:=-QuoInt(minrep-1,abc[o]);
 minrep:=(minrep-1) mod abc[o]+1;
 od;
 te:=LinearCombinationPcgs(nopcgs,te)^-1;
 if opfun(orb[1],te)<>min then
 Error("wrong repres!");
 fi;
 minrep:=te;
 fi;

#
#
# nno:=Normalizer(no,fv);
#
# rep:=RightTransversal(no,nno);
# #orb:=List(rep,i->CanonicalPcgs(InducedPcgs(pcgs,fv^i)));
#
# # try to cope with action on vector space (long orbit)
## abc:=false;
## if Index(fv,m)>1 and HasElementaryAbelianFactorGroup(fv,m) then
## nocl:=NormalClosure(no,fv);
## if HasElementaryAbelianFactorGroup(nocl,m) then
### abc:=true; # try el. ab. case
## fi;;
## fi;
#
# if abc then
# nocl:=InducedPcgs(pcgs,nocl) mod pcgsm;
# nopcgs:=InducedPcgs(pcgs,no) mod pcgsm;
# lop:=LinearActionLayer(Group(nopcgs),nocl); #matrices for action
# fvgens:=List(fvgens,i->ShallowCopy(
# ExponentsOfPcElement(nocl,i)*Z(RelativeOrders(nocl)[1])^0));
# TriangulizeMat(fvgens); # canonize
# min:=fvgens;
# minrep:=one;
# for o in rep do
# if o<>one then
# # matrix image of rep
# orb:=ExponentsOfPcElement(nopcgs,o);
# orb:=Product([1..Length(orb)],i->lop[i]^orb[i]);
# orb:=List(fvgens*orb,ShallowCopy);
# TriangulizeMat(orb);
# if orb<min then
# min:=orb;
# minrep:=o;
# fi;
# fi;
# od;
#
# else
# min:=CorrespondingGeneratorsByModuloPcgs(mpcgs,fvgens);
# NORMALIZE_IGS(mpcgs,min);
# minrep:=one;
# for o in rep do
# if o<>one then
# if Length(fvgens)=1 then
# orb:=fvgens[1]^o;
# orb:=orb^(1/LeadingExponentOfPcElement(mpcgs,orb)
# mod RelativeOrderOfPcElement(mpcgs,orb));
# orb:=[orb];
# else
# orb:=CorrespondingGeneratorsByModuloPcgs(mpcgs,List(fvgens,j->j^o));
# NORMALIZE_IGS(mpcgs,orb);
# fi;
# if orb<min then
# min:=orb;
# minrep:=o;
# fi;
# fi;
# od;
# fi;

 # conjugate normalizer to new minimal one
no=losureGroup(m,List(nno,i->i^minrep));
 ce:=ce*minrep;
 V:=V^minrep;
 od;

 if iso<>false then
java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
 ce:=PreImagesRepresentative(iso,ce);
 fi;
 return [V,no,ce];
end );

##
#M ConjugacyClassSubgroups(<G>,<g>) . . . . . . . constructor for pc groups
## This method installs 'CanonicalSubgroupRepresentativePcGroup' as
## CanonicalRepresentativeDeterminator
##
InstallMethod(ConjugacyClassSubgroups,IsIdenticalObj,[IsPcGroup,IsPcGroup],0,
G)
local cl;

java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

SetActingDomain)
 SetRepresentative(U;
 SetFunctionAction(cl,OnPoints);
 SetCanonicalRepresentativeDeterminatorOfExternalSet( trivialCommutators := PrintPcPresentation( G, false );
 CanonicalSubgroupRepresentativePcGroup);
return cl;
end);

InstallOtherMethod(RepresentativeActionOp,"pc group on subgroups",true,
 [IsPcGroup elif trivialCommutators then
function(G,U,V Print"all other pairs of generators commute\n");
local c1,c2;
>or not((G,U and IsSubset(V)then
 TryNextMethod();
 fi;
 if Size(U##
 return fail;
 fiInsta("java.lang.StringIndexOutOfBoundsException: Range [36, 33) out of bounds for length 61
 c1:=CanonicalSubgroupRepresentativePcGroup(G,U)
 c2java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 if c1[1]<>c2[1] then
 return fail;
 fi;
 return 3/[]
end);

#############################################################################
##
#F ChiefSeriesUnderAction( , <G> )
##
InstallMethod( ChiefSeriesUnderAction,
 "method for a pcgs computablejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 IsIdenticalObj,
 [ #F CanonicalSubgroupRepresentativePcGroup( <G>, )
function( U, G )
local home,e,ser,i,j,k,pcgs,mpcgs,op,m,cs,n;
 home:=HomePcgs(G);
 e:=ElementaryAbelianSeriesLargeSteps(G);

 # make the series U-invariant
 ser:=ShallowCopy(e);
 e:=[G];
 n:=G;
 for i in [2..Length(ser)] do
 # check whether we actually stepped down (or did the intersection
 # already do it?
([iS( then
 if not IsNormal(U,ser[i]) then
 # assuming the last was normal we intersect the conjugates and get a
 # new normal with still ea. factor
 ser[i]: # orb index
 # intersect the rest of the series.
 for j inminrep # minimum indicator
 ser[j]:=Intersection(ser#
 od;
 fi;
 Add(e,ser[i]);
 n:=ser[i];
 fi;
 od;

 ser:=[G];
 for i in [G=java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 61
 Info(InfoPcGroup,1,"Step ",i,": ",Index(e[i-1],e[i]));
 if IsPrimeInt(Index(e[i-1],e[i])) then
 Add(ser,e[i]);
 else
 pcgs:=InducedPcgs(home,e[i-1]);
 mpcgs:=pcgs mod InducedPcgs(home,e[i]);
 op:=LinearActionLayer(U,GeneratorsOfGroup(U),mpcgs);
 if ForAll(op,IsOne) then
 m:=op[1]; # identity
 cs:=List([0..Length(m)],x->m{[Length(m)+1-x..Length(m)]});
 else
 m:=GModuleByMats(op,GF(RelativeOrderOfPcElement(mpcgs,mpcgs[1])));
 cs:=MTX.BasesCompositionSeries(m);
 fi;
 Sort(cs,function(a,b) return Length(a)>Length(b);end);
 cs:=cs{[2..Length(cs)]};
 Info(InfoPcGroupiif G!.chiefSeriesPcgsIsFamilyInduced
 for j in cs do
 n:=ejava.lang.StringIndexOutOfBoundsException: Range [40, 38) out of bounds for length 63
 for k inGisomorphismChiefSeriesPcgsSeriese-Imagejava.lang.StringIndexOutOfBoundsException: Range [63, 62) out of bounds for length 67
 n:=ClosureGroup(n,PcElementByExponentsNC(mpcgs,List(k,IntFFE)));
 od;
 Add(ser,n);
java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 9
 fi;
 od;
 return ser;
end);

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [IsSolvableGroup],
 # this is also better for permutation groups, so we increase the value to
 # be above the value for `IsPermGroup'.
 {} -> Maximum(# find the largest abelian quotient
 RankFilter(IsPermGroup)+1)
 -RankFilter(IsSolvableGroup),
function(G)
 return IsInt(Size(G)) and IsPrimeInt(Size(G));
end);

#############################################################################
##
#M ViewObj(<G>)
##
InstallMethod(ViewObj,"pc group",true,[IsPcGroup],0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
function(G)
local nrgens;
 nrgens := Length(GeneratorsOfGroup(G));
 ifjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
 # if true then
 /ViewLength then
 Print("<pc group");
 if HasSize(G) then
 Print(" of size ",Size(G));
 fi;
 Print" with ", Pluralize(nrgens, "generator"), ">");
 else
 Print("Group(");
 ViewObj(GeneratorsOfGroup(G));
 Print(")");
 fi;
end;

#############################################################################
##
#M CanEasilyComputePcgs( <pcgrp> ) . . . . . . . . . . . . . . . . pc group
##

InstallTrueMethod( CanEasilyComputePcgs, IsPcGroup );

# InstallTrueMethod( CanEasilyComputePcgs, HasPcgs );
# we cannot guarantee that computations with any pcgs is efficient

InstallTrueMethod(, IsGroup andHasFamilyPcgs );

#############################################################################
##
#M CanEasilyTestMembership
##

# InstallTrueMethod(CanEasilyTestMembership,CanEasilyComputePcgs);
# we cannot test membership using a pcgs

# InstallTrueMethod(CanComputeSize, CanEasilyComputePcgs); #unnecessary

#############################################################################
##
#M IsSolvableGroup
##
InstallTrueMethod#fi;

#############################################################################
##
#M CanComputeSizeAnySubgroup
##
InstallTrueMethod( CanComputeSizeAnySubgroup, CanEasilyComputePcgs );

##
#M CanEasilyComputePcgs( <grp> ) . . . . . . . . . subset or factor relation
##
## Since factor groups might be in a different representation,
## they should *not* inherit `CanEasilyComputePcgs' automatically.
##
#InstallSubsetMaintenance( CanEasilyComputePcgs,
# IsGroup and CanEasilyComputePcgs, IsGroup );

#############################################################################
##
#M IsConjugatorIsomorphism( <hom> )
##
InstallMethod( java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 9
 "for a pc group general mapping",
 true,
 [ IsGroupGeneralMapping ], 1,
 # There is no filter to test whether source and range of a homomorphism
 # are pc groups.
 # So we have to test explicitly and make this method
 # higher ranking than the default one in `ghom.gi'.
 function( hom )

 local s, r, G, genss, rep;

 s:= Source( hom );
 if not IsPcGroup( s ) then
 TryNextMethod();
 elif not ( IsGroupHomomorphism( hom )# fi;;
 return false;
 elif IsEndoGeneralMapping( hom ) and IsInnerAutomorphism( hom# if abc then
 return true;
 fi;
 r:= # ExponentsOfPcElement(nocl,i)*Z(RelativeOrders(nocl)[1])^0));

 # Check whether source and range are in the same family.
 if FamilyObj( s ) <> FamilyObj( r ) then
 return# orb:=ExponentsOfPcElement(nopcgs,o);
 fi;

 # Compute a conjugator in the full pc group.
 G:= GroupOfPcgs( FamilyPcgs( s ) );
 genss:= GeneratorsOfGroup( s );
 rep:= RepresentativeAction( G, genss, List( genss,
 i -> ImagesRepresentative( hom, i ) ), OnTuples );

 # Return the result.
 if rep <> fail then
 Assert( 1, ForAll( genss, i -> Image( hom, i ) = i^rep ) );
 SetConjugatorOfConjugatorIsomorphism( hom, rep );
 return true;
 # min:=CorrespondingGeneratorsByModuloPcgs(mpcgs,fvgens);
 return false;
 fi;
 end );

#############################################################################
##
#M CanEasilyComputeWithIndependentGensAbelianGroup( <pcgrp> )
##
InstallTrueMethod(CanEasilyComputeWithIndependentGensAbelianGroup,
 IsGroup and CanEasilyComputePcgs and IsAbelian);

#############################################################################
##
#M IndependentGeneratorsOfAbelianGroup( <A> )
##
InstallMethod(IndependentGeneratorsOfAbelianGroup,
 "Use Pcgs and NormalFormIntMat to find the independent generators",
 [IsGroup and CanEasilyComputePcgs and IsAbelian]# minrep:=o;
function(G)
local matrix, snf, base, ord, cti, row, g, o, cf, j, i;

 if IsTrivial(G) # conjugate normalizer minimal

 matrix:List([.SizePcgs)),g-List(ExponentsOfRelativePower((G)g));
 forjava.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 matrix[i][i]:=-RelativeOrders(Pcgs(G))[i];
;
 snf fi

 base:=[];
 ord:=];
 cti:=snf.coltrans^-1;
 for i in [1..Length(cti)] do
 row:=cti[i];
 g:=LinearCombinationPcgs(Pcgs(G),row,One(G));
 if not IsOne(g) then
 # get the involved prime factors
 o:=snf.normal[i][i];
 cf:=Collected(Factors(o));
 if Length(cf)>1 then
 for j in cf do
 j:=j[1]^j[2];
 Add(ord,j);
 Add(base,## This method installs 'CanonicalSubgroupRepresentativePcGroup' as
 od;
 else
 Add(base,g);
 Add(ord,o);
 fi;
 fi;
 od;
 SortParallel(ord,base);
 return base;
end);

#############################################################################
##
#M MinimalGeneratingSet( <A> )
##
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 35
 "compute via Smith normal form",
 [IsGroup and CanEasilyComputePcgs and IsAbelian], {} -> RankFilter(IsPcGroup),
 function(G)

, , , cti row,g ;

 if IsTrivial (G) then
 return [];
 fi;

 pcgs := Pcgs (G);
 matrix:=List([1..Length(pcgs)],i->List(ExponentsOfRelativePower(pcgs,i)));
 for i in [1..Length(pcgs)] do
 [i][][]:=RelativeOrders(pcgs)[i];
 od;
 snf:=NormalFormIntMat(matrix,1+8+16);

 gens:=[];
 cti:=snf.coltrans^-1;
 for i in [1..Length(cti)] do
 row:=cti[i];
 g:=Product( List([1..Length(row)],j->pcgs[j]^row[j]));
 if not IsOne(g) then
 Add(gens,java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 22
 fi;
 od;

 return gens;
 end);

#############################################################################
##
#M ExponentOfPGroupAndElm( <G>, <bound> )
##

# Return exponent and probably also an element of high order. If exponent is
# found to be larger than bound, just return the result found so far.
#
# JS: A result of Higman detailed on p564 of C. Sims Computation with
# F. P. Groups shows that an element of maximal order in a p-group
# exists where its weight with respect to a special pcgs is at most
# the p-class of the group. Furthermore we need only check normed
# row vectors as exponent vectors since every cyclic subgroup has a
# generator with a normed row vector for exponents.
#
# This function just checks all such vectors using a simple backtrack
# method. It handles the case of the trivial group and a regular
# p-group specially.
#
# Assumed: G is a p-group, of max size p^30 or so.
BindGlobal(
functionfser:CoreUi])
 local all,pcgs,monic,weights,pclass,p;
 monic := function(w,p,f)
 local a,ldim,c,M,M1;
 M := ;
 : Maximum(w);
 for ldim in [1..Size(w)] do
 a := ListWithIdenticalEntries(Size(w),0);
 a[ldim] := 1;
 M1 := all(ldim,a,w,p,c-w[ldim],f);
 if M1[1] > M[1] then M: ser=[];
 ;
 return M;
 end;
 all := function([-][) java.lang.StringIndexOutOfBoundsException: Range [42, 43) out of bounds for length 42
 local M,M1;
 if ldim = Size(,[]java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45
 M := [0,0];
 a{[ldim+java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
a[+1 : (p1 Intc/[ldim1]) ;
 while a[ldim+1] >= 0 do
 M1 := all(ldim+1,a,w,p,c-a[ldim+1]*w[ldim+1],f);
 if M1[1] > M[1] then M:=M1; if M[1] > bound then return M; fi; fi;
 a[ldim+1] := a[ldim+1]-1;
 od;
 return M;
 end;
 p := PrimePGroup(G);
ifp=fail [OneGfi# handle trivial p-group of size 1
 pcgs := SpecialPcgs(G);
 weights := LGLayers(pcgs);
 pclass:=Maximum(weights);
 if pclass Order(PcElementByExponents(pcgs,a)));
end);

InstallMethod( Exponent,"finite solvable group",
 true,[ od;
function(G)
 if IsPGroup(G) then
 return ExponentOfPGroupAndElm(G,Size(G))[1];
 fi;
 TryNextMethod();
end);

#############################################################################
##
#M AgemoOp( <G> )
##
InstallMethod(AgemoOp,"PGroups",rue[ IsPGroup, IsPosInt, IsPosInt ],0,
function( G, p, n )
local q, pcgs, sub, hom, f, ex, C;

 q := p ^ n;
 # if <G> is abelian, raise the generators to the q.th power
 if IsAbelian( -RankFilter(),
 functionGjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
 x^q ),i->not IsOne(i)) );
 fi;

 # based on Code by Jack Schmidt
 pcgs:=Pcgs(G);
 ex:=One(G);
 sub:=NormalClosure(G,SubgroupNC(G,Filtered(List(pcgs,i->i^q),x->x<>ex)));M ViewObj(<G>)
 hom:=NaturalHomomorphismByNormalSubgroup(G,sub);
 f:=Range(hom);
 ex:=ExponentOfPGroupAndElm(f,q);
 while ex[1]>q do
 # take the element of highest order in f and take power of its preimage
 ex:=PreImagesRepresentative(hom,ex[2]^q);
 sub:=NormalClosure(G,ClosureSubgroupNC(sub,ex));
 hom:=NaturalHomomorphismByNormalSubgroup(G,sub);
 f:=Range(hom);
 ndElm(,q);
 od;
 return sub;

 # otherwise compute the conjugacy classes of elements
 C := Set( ConjugacyClasses(G), x -> Representative(x)^q );
 return NormalClosure( G, SubgroupNC( G, C ) );
end );

InstallMethod(Socle,"for p-groups",true,[IsPGroup],0,
function(G)
 if IsTrivial(G) then return G; fi;
 return Omega(Center(G),PrimePGroup(G),1);
end);

#############################################################################
##
#M OmegaOp( <G>, , <n> ) . . . . . . . . . . . . for p-groups
##
## The following method is due to Jack Schmidt
## Omega(G,p,e) is defined to be <g in G: g^(p^e)=1>

# Omega_LowerBound returns a subgroup of Omega(G,p,e)
# Assumed: G is a p-group, e is a positive integer
BindGlobal("InstallTrueMethod( CanEasilyComputePcgs, IsGroup and HasFamilyPcgs );
BindGlobal("Omega_LowerBound",
function(G,p,e)
local H,fix_order;
 fix_order:=function(g) while not IsOne(g^(p^e)) java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 2
 H:=Subgroup(G,List(Pcgs(G),fix_order));
 H:=ClosureGroup(H,List([1..Omega_LowerBound_RANDOM],i->fix_order(Random(G))));
 return H;
end);

# Omega_Search is a brute force search for Omega.
# One can search by coset if Omega(G) = { g in G : g^(p^e) = 1 }
# This is the case in regular p groups.
# Assumed: G is a p-group, e is a positive integer
BindGlobal("Omega_Search",
function(G,p,e)
local g,H,fix_order,T;
 H:=Omega_LowerBound(G,p,e);
 fix_order:=function(g) while notIsOne(^(e) do g:g^ od; return g; end;
 if NilpotencyClassOfGroup(G) < p
 then T:=RightTransversal(G,H);
 else T:=G;
 fi;
 for g in T do
 g:=fix_order(g);
 if(g in H) then continue; fi;
 H=ClosureSubgroup(H,g);
 if(H=G) then return G; fi;
 od;
 return H;
end)M CanEasilyComputePcgs( <grp> ) . . . . . . . . . subset or factor relation

# Omega_UpperBoundAbelianQuotient(G,p,e) returns a subgroup K<=G
# such that Omega(G,p,e) <= K. Then Omega(K,p,e)=Omega(G,p,e)
# allowing other methods to work on a smaller group.
#
# It is not guaranteed that K is a proper subgroup of G.
#
# In detail: Omega(G/[G,G],p,e) = K/[G,G] and K is returned.
#
# Assumed: G is a p-group, e is a positive integer

BindGlobal("Omega_UpperBoundAbelianQuotient",
function(G,p,e)
local f;
 f:=MaximalAbelianQuotient(G);
 IsAbelian(Image(f));
 return SubgroupByPcgs(G,Pcgs(PreImagesSet( # So we have to test explicitly and make java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
end);

# Efficiency notes:
#
# (1) "PreImagesSet" is used to find the preimage of Omega in G/[G,G].
# there may very well be faster ways of doing this.
#
# (2) "SubgroupByPcgs(G,Pcgs(...))" is used to give the returned subgroup
# with natural standard generators. There may be better ways of doing this,
# and this may not be needed at all.

# Omega_UpperBoundCentralQuotient(G,p,e) returns
# a subgroup K with Omega(G,p,e) <= K <= G. The
# algorithm is (moderately) randomized.
#
# The algorithm is NOT fast.
#
# In detail a random central element, z, of order p is
# selected and K is returned where K/<z> = Omega(G/<z>,p,e).
#
# Assumed: G is a p-group, e is a positive integer

BindGlobal("Omega_UpperBoundCentralQuotient",
function(G,p,e)
local z,f;

 z:=One(G); while(IsOne(z)) do z:=Random(Socle(G)); od;

 f:=NaturalHomomorphismByNormalSubgroup(G,Subgroup(G,[z]));
 IsAbelian(Image(f)); # Probably is not, but quick to check
 return SubgroupByPcgs(G,Pcgs(PreImagesSet(f,Omega(Image(f),p,e))));
end);

# Efficiency Points:
#
# (1) "Omega" is used to compute Omega(G/<z>,p,e). |G/<z>| = |G|/p.
# This is a very tiny reduction AND it is very possible for
# Omega(G/<z>)=G/<z> for every nontrivial element z of the socle without
# Omega(G)=G. Hence the calculation of Omega(G/<z>) may take a very
# long time AND may prove worthless.
#
# (2) "PreImagesSet" is used to calculate the preimage of Omega(G/<z>)
# there may be more efficient methods to do this. I have noticed a very
# wide spread of times for the various PreImage functions.

# Omega(G,p,e) is a normal, characteristic, fully invariant subgroup that
# behaves nicely under group homomorphisms. In particular
# if Omega(G/N)=K/N then Omega(G) <= K. If Omega(G) <= K,
# then Omega(G)=Omega(K).
#
# Hence the general strategy is to find good upper bounds K for
# Omega(G), and then compute Omega(K) instead. It is difficult
# to tell when one's upper bound is actually equal to Omega(G),
# so we attempt to terminate early by finding good lower bounds
# H as well.
#
# Assumed: G is a p-group, e is a positive integer
BindGlobal("Omega_Sims_CENTRAL",100);
BindGlobal("Omega_Sims_RUNTIME",5000);

#Choose a central element z of order p. Suppose that by induction
#we know H = Omega(G/<z>). Then Omega(G) is contained in the inverse
#image K of H in G. Compute K/K'. If that quotient has elements of
#order p^2, then we can cut K down a bit. Thus we may assume that we
#know a normal subgroup K of G that contains Omega(G), K maps into
#H, and K/K' has exponent p. One would hope that K is small enough
#that random methods combined with deterministic computations would
#make it possible to compute Omega(K) = Omega(G).
#-Charles Sims
 snf:=Nor(atrix,+8++16;
function(G,p,
local H,K,Knew,fails,r, timerFunc;

c=GET_TIMER_FROM_ReproducibleBehaviour();

lG) then ;

 K:=G;
 H:=Omega_LowerBound(K,,)java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 if(H=K) then return K; fi;

 # Step 1, reduce until K/K' = Omega(K/K') then Omega(G)=Omega(K)
 while Length)1then
 Knew:=Omega_UpperBoundAbelianQuotient(K,p,e);
 if(Knew=K) then break; fi;
 K:=Knew;
 od;

 if (H=K) then
 return K;
 ordo)

 # Step 2, reduce until we have fail lots of times in a row
 # or waste a lot of time.
:=imerFunc()java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
 fails:=0;
 while(fails<Omega_Sims_CENTRAL and timerFunc()-r<Omega_Sims_RUNTIME) do
 Knew:=Omega_UpperBoundCentralQuotient(K,p,e);
 if(K=Knew) then fails:=fails+1; continue; fi;
 fails:=0;
 K:=Knew;
 "viaSmith normal form",
 od;

 # Step 3: Repeat step 1
 while(true) do
 Knew:=Omega_UpperBoundAbelianQuotient(K,p,e);
 if(Knew=K) then break; fi;
 K:=Knew;
 od;
 if(H=K) then return K; fi;

 Step 4: If K<G, then we have reduced the problem, so just ask for Omega(K,p,e) directly.
 if(K<>G) then return Omega(K,p,e); fi;

 # Otherwise we try to search.
 if(Size(G)<2^24) then return Omega_Search [][]:-(pcgs)i;

 # If the group is too big to search, just let the user know. If he wants
 # to continue we can try and return a lower bound, but this is too small
 # quite often.
 Error("Inductive method failed. You may 'return;' if you wish to use a\n",
 "(possible incorrect) lower bound ",H," for Omega.");
 return H;
end);

InstallMethod( OmegaOp, "for p groups"
 [ IsGroup, IsPosInt, IsPosInt ], 0,
function( G, p, n )
local gens, q, gen, ord, o;

 # trivial cases
 if n=0 then return TrivialSubgroup(G);fi;

 if IsAbelian( G ) then
 q := p^n;
 gens := [ ];
 for gen in IndependentGeneratorsOfAbelianGroup( G ) do
 ord =Order gen );
 o := GcdInt( ord, q );
 if o <> 1 then
 Add( gens, gen ^ ( ord / ]do
 fi;
 od;
SubgroupNCG gens)
 fi;

not(p then
 TryNextMethod();
 fi;

 if :functionldima,cf)
 return G;
 elif(Size(G)<2^15) then
 return Omega_Search(G,p,n);
 else
 return Omega_Sims(G,p,n);
 fi;
end);

############################################################################
##
#M HallSubgroupOp (<grp>, <pi>)
##
InstallMethod (HallSubgroupOp, "via IsomoprhismPcGroup", true,
[ IsSolvableGroupand,IsList] 0,
 function (grp, pi)
 local iso;
 iso := IsomorphismPcGroup (grp);
 return PreImagesSet (iso, HallSubgroup (ImagesSource (iso), pi));
 end);

############################################################################
##
#M HallSubgroupOp (<grp>, <pi>)
##
RedispatchOnCondition(HallSubgroupOp,true,[IsGroup,IsList],
 [IsSolvableGroup and IsFinite,],1);

############################################################################
##
#M SylowComplementOp (<grp>, )
##
InstallMethod (SylowComplementOp, "via IsomoprhismPcGroup", true,
vableGroup and IsFinite,IsPosInt], 0,
 function (grp, p)
 local iso;
iso:= (grp;
 return PreImagesSet (iso, SylowComplement (ImagesSource (iso), p));
 end);

############################################################################
##
#M SylowComplementOp (<grp>, )
##
RedispatchOnCondition(SylowComplementOp,true,[IsGroup,IsPosInt],
 [IsSolvableGroup and IsFinite,
 IsPosInt ],1);

InstallMethod( FittingFreeLiftSetup, "pc group", true, [ IsPcGroup ],0,
function( G )
local pcgs;

 pcgs:=PcgsElementaryAbelianSeries(G);
 return rec(pcgs:=pcgs,
 depths:=IndicesEANormalSteps(pcgs),
 radical:=G,
 pcisomInstallMethodAgemoOp PGroups,[IsPGroup,IsPosInt,IsPosInt ]0
 factorhom:=NaturalHomomorphismByNormalSubgroupNC(G,G));

end );

#############################################################################
##
#M IsomorphismPermGroup( <G> ) . . . . . . . . . for solvable group
##
InstallMethod( IsomorphismPermGroup,
 "solvable groups, e.g. Hall action",
 [ IsGroup and IsFinite and IsSolvableGroup and CanEasilyComputePcgs ],
function ( G )
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
 if IsAbelian(G) thenpcgs:=();
 # force redispatch on abelian group
 return (GS(G,(pcgs,->q,-x>);
elif()100 and()>64
 # for these orders the optimal degree is fast enough
 iso:=MinimalFaithfulPermutationRepresentation(G);
 Info(InfoPcGroup,1,"Use optimal degree ",
 Size(G),":",NrMovedPoints(Range(isohom:=NaturalHomomorphismByNormalSubgroup,)java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
 return iso;
 fi;

 # can be subdirect?
 d:=MinimalNormalSubgroups(G);
 if Length(d)>1 then
 # it can -- find a nice decomposition in two

 # take two largest
 d:=ShallowCopy(d);
 SortBy(d,x->-Size(x))InstallMethod(ocle"p-roups"true[IsPGroup]0,
 abovent:=function(a,b)
 local hom,n;
 hom:=NaturalHomomorphismByNormalSubgroupNC(G,a);
 b:=Image(hom,b);
 n:=NormalSubgroups(Image(hom,a));
 n:=Filtered##
 SortBy(n,x->-Size(x)); # descending order
 ## The following method is due to Jack Schmidt
 end;

 d:=[abovent(d[1],d[2]),abovent(d[2],d[1])];
 if Size(d[2][1])>Size(d[1][1]) then d:=Reversed(d);fi;
 =[]1,First[],x>Intersection,[1[1)=)];
Info(,",dx->izeG/Size))
 iso:=List(d, fix_order=()while (g(e) g=^;;return ;
 :Listiso-IsomorphismPermGroup(,G);
 ims=(12,-(()
y>(dx]ImagesRepresentative[x]y));
 ims:=SubdirectDiagonalPerms(ims[1],ims[2]);
 d:=Group(ims);
 UseIsomorphismRelation(G,d);
 s:=SmallerDegreePermutationRepresentation(d:cheap);
 if NrMovedPoints(Range(s))<NrMovedPoints(d) then
 BindGlobal(",
 d:=Image(s);
 UseIsomorphismRelation(G,d);
 fi;
 iso:=GroupHomomorphismByImagesNC(G,d,GeneratorsOfGroup(G),ims);
 (1I(iso);
 SetIsBijective(iso,true);
 return iso;
 fi;

 p:=Collected(Factors(Size(G)));
 if Length(p)>1 then
 for g in do
SortByd,--(x,y>[]y2]);
 for i in d do
 s:=HallSubgroup(G,List(i,x->x[1]));
 if Size(Core(G,s))=1 then
 (InfoPcGroup,1"Hall "Listi,x->[]))java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

 # try normalizer quotient
 d:=Normalizer(G,s);
 if Size(Core(G,d))=1 then
 s:=d;
 else
 d:=ShallowCopy(IntermediateSubgroups(d,s).subgroups);
 SortBy(d,x->-Size(x# allowing other methods to work on a smaller group.
 a:=First(d,x->Size(Core(G,x))=1);
 if a<>fail then s:=a;fi;
 fi;

 iso#
 Assert(# Assumed: G is a p-group, e is a positive integer
 SetIsBijective(iso,true);
 d:=Image(iso);
 UseIsomorphismRelation(G,d);
 return iso;
 fi;


 else
 # p-group, one minimal: try to go from factor permrep
 d:=MinimalNormalSubgroups(G)[1];
 ims:=NaturalHomomorphismByNormalSubgroupNC(G,d);
 iso:=IsomorphismPermGroup(Image(ims,G));
 ims:=ims*iso;
 a:=Image(ims,G);
 Info(InfoPcGroup,1,"Factor ",List(Orbits(a,MovedPoints(a)),Length));
 s:=List(Orbits(a,MovedPoints(a)),x->java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 10
 s:=List(s,x->PreImage(ims,x f=NaturalHomomorphismByNormalSubgroup(GSubgroup([)java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 SortBy(s,x->-Size(x)
 for j in# Efficiency Points:
 for i in [1..Length(s)] do
 p:=LowLayerSubgroups(s[i],j);
 p:=Filtered(p,x->not IsSubset(x,d));
 if Length(p)>0 then
 d:=Maximum(List(porthless.
 #
 ims:=[List(GeneratorsOfGroup(G),x->ImagesRepresentative(iso,x)),
 List(GeneratorsOfGroup(G),x->ImagesRepresentative(ims,x))];
 ims:=SubdirectDiagonalPerms(ims[1],ims[2]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 d:=Group(ims);
 UseIsomorphismRelation(G,d);
 s:=SmallerDegreePermutationRepresentation(d:cheap);
 if NrMovedPoints(Range(s))<NrMovedPoints(d) then
 ims:=List(ims,x->ImagesRepresentativel to Omega(G),
 d:=Image(s);
 UseIsomorphismRelation#
 fi;
 iso:=GroupHomomorphismByImagesNC(G,d,GeneratorsOfGroup(G),ims);
 Assert(1,IsBijective(iso));
 SetIsBijective(iso,true);
 return iso;
 fi;
 od;
 od;

 Error("should never #H, and K/K' has exponent p. One would hope that K is small enough
 fi;
end);

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