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############################################################################
##
#W grppcext.gi GAP library Bettina Eick
##
#Y Copyright (C) 1997, Lehrstuhl D für Mathematik, RWTH Aachen, Germany
#Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
#############################################################################
##
#F FpGroupPcGroupSQ( G ). . . . . . . . .relators according to sq-algorithmus
##
InstallGlobalFunction( FpGroupPcGroupSQ, function( G )
local F, f, g, n, rels, i, j, w, v, p, k;
F := FreeGroup(IsSyllableWordsFamily, Length(Pcgs(G)) );
f := GeneratorsOfGroup( F );
g := Pcgs( G );
n := Length( g );
rels := List( [1..n], x -> List( [1..x], y -> false ) );
for i in [1..n] do
for j in [1..i-1] do
w := f[j]^-1 * f[i] * f[j];
v := ExponentsOfPcElement( g, g[j]^-1 * g[i] * g[j] );
for k in Reversed( [1..n] ) do
w := w * f[k]^(-v[k]);
od;
rels[i][j] := w;
od;
p := RelativeOrderOfPcElement( g, g[i] );
w := f[i]^p;
v := ExponentsOfPcElement( g, g[i]^p );
for k in Reversed( [1..n] ) do
w := w * f[k]^(-v[k]);
od;
rels[i][i] := w;
od;
return rec( group := F, relators := Concatenation( rels ) );
end );
#############################################################################
##
#F MappedPcElement( elm, pcgs, list )
##
InstallGlobalFunction(MappedPcElement,function( elm, pcgs, list )
local vec, new, i;
if Length( list ) = 0 then return fail; fi;
vec := ExponentsOfPcElement( pcgs, elm );
if Length( list ) < Length( vec ) then return fail; fi;
new := false;
for i in [1..Length(vec)] do
if vec[i]>0 then
if new=false then
new := list[i]^vec[i];
else
new := new * list[i]^vec[i];
fi;
fi;
od;
if new=false then
new:=One(list[1]);
fi;
return new;
end);
#############################################################################
##
#F TracedPointPcElement( elm, pcgs, imgs,pt )
##
InstallGlobalFunction(TracedPointPcElement,function( elm, pcgs, list,pt )
local vec, i,j;
if Length( list ) = 0 then return pt; fi;
vec := ExponentsOfPcElement( pcgs, elm );
if Length( list ) < Length( vec ) then return fail; fi;
for i in [1..Length(vec)] do
if vec[i]>0 then
for j in [1..vec[i]] do
pt:=pt^list[i];
od;
fi;
od;
return pt;
end);
#############################################################################
##
#F ExtensionSQ( C, G, M, c )
##
## If <c> is zero, construct the split extension of <G> and <M>
##
InstallGlobalFunction( ExtensionSQ, function( C, G, M, c )
local field, d, n, rels, i, j, w, p, k, l, v, F, m, relators, H, orders,
Mgens;
# construct module generators
field := M.field;
Mgens := M.generators;
d := M.dimension;
orders := List([1..d], x -> Characteristic(M.field));
if Length(Mgens) = 0 then
return AbelianGroup( orders );
fi;
n := Length(Pcgs( G ));
# add tails to presentation
if c = 0 then
rels := ShallowCopy( C.relators );
else
rels := [];
for i in [ 1 .. n ] do
rels[i] := [];
for j in [ 1 .. i ] do
if C.relators[i][j] = 0 then
w := [];
else
w := ShallowCopy(C.relators[i][j]);
fi;
p := (i^2-i)/2 + j - 1;
for k in [ 1 .. d ] do
l := c[p*d+k];
if not IsZero( l ) then
Add( w, n+k );
Add( w, IntFFE(l) );
fi;
od;
if 0 = Length(w) then
w := 0;
fi;
rels[i][j] := w;
od;
od;
fi;
# add module
for j in [ 1 .. d ] do
rels[n+j] := [];
for i in [ 1 .. j-1 ] do
rels[n+j][n+i] := [ n+j, 1 ];
od;
rels[n+j][n+j] := 0;
od;
# add operation of <G> on module
for i in [ 1 .. n ] do
for j in [ 1 .. d ] do
v := Mgens[i][j];
w := [];
for k in [ 1 .. d ] do
l := v[k];
if not IsZero( l ) then
Add( w, n+k );
Add( w, IntFFE(l) );
fi;
od;
rels[n+j][i] := w;
od;
od;
orders := Concatenation( C.orders, orders );
# create extension as fp group
F := FreeGroup(IsSyllableWordsFamily, n+d );
m := GeneratorsOfGroup( F );
# and construct new presentation from collector
relators := [];
for i in [ 1 .. d+n ] do
for j in [ i .. d+n ] do
if i = j then
w := m[i]^orders[i];
else
w := m[j]^m[i];
fi;
v := rels[j][i];
if 0 <> v then
for k in [ Length(v)-1, Length(v)-3 .. 1 ] do
w := w * m[v[k]]^(-v[k+1]);
od;
fi;
Add( relators, w );
od;
od;
# Error("A");
H := PcGroupFpGroup( F / relators );
SetModuleOfExtension( H, Subgroup(H, Pcgs(H){[n+1..n+d]} ) );
return H;
end );
#############################################################################
##
#F FastExtSQ( G, M, c,check )
##
##
BindGlobal( "FastExtSQ", function( G, M, c,check )
local field, d, n, i, j, w, p, k, l, v, F, H, orders,
Mgens,pcgs,z,fam,col,exp;
pcgs:=Pcgs(G);
# construct module generators
field := M.field;
z:=Zero(field);
Mgens := M.generators;
if Length(Mgens) = 0 then
return AbelianGroup( List([1..M.dimension],
x -> Characteristic(M.field)));
fi;
d := Length(Mgens[1]);
n := Length(pcgs);
F:=FreeGroup(IsSyllableWordsFamily,d+n);
fam:=FamilyObj(One(F));
orders := Concatenation( RelativeOrders(pcgs), List( [1..d],
x -> Characteristic( field ) ) );
col:=SingleCollector(GeneratorsOfGroup(F),orders);
for i in [1..n] do
for j in [1..i] do
if i=j then
exp:=ExponentsOfRelativePower(pcgs,i);
else
exp:=ExponentsOfConjugate(pcgs,i,j);
fi;
w:=[];
# start at j -- there cannot be earlier entries.
for k in [j..n] do
if exp[k]<>0 then
Add(w,k);
Add(w,exp[k]);
fi;
od;
if not IsInt(c) then # add cocycle info
p := (i^2-i)/2 + j - 1;
for k in [ 1 .. d ] do
l := c[p*d+k];
if l <> z then
Add( w, n+k );
Add( w, IntFFE(l) );
fi;
od;
fi;
if Length(w)>0 then # other relators are considered trivial
if i=j then
w:=ObjByExtRep(fam,w);
SetPower(col,i,w);
elif w<>[i,1] then
w:=ObjByExtRep(fam,w);
SetConjugate(col,i,j,w);
fi;
fi;
od;
od;
# module relations do not need to be written down -- they are all
# trivial
# add operation of <G> on module
for i in [ 1 .. n ] do
for j in [ 1 .. d ] do
v := Mgens[i][j];
w := [];
for k in [ 1 .. d ] do
l := v[k];
if l <> z then
Add( w, n+k );
Add( w, IntFFE(l) );
fi;
od;
if Length(w)>0 and w<>[n+j,1] then
w:=ObjByExtRep(fam,w);
SetConjugate(col,n+j,i,w);
fi;
od;
od;
if check then
H := GroupByRws(col);
else
H := GroupByRwsNC(col);
fi;
SetModuleOfExtension( H, Subgroup(H, Pcgs(H){[n+1..n+d]} ) );
return H;
end );
#############################################################################
##
#M Extension( G, M, c )
##
InstallMethod( Extension, "generic method for pc groups", true,
[ CanEasilyComputePcgs, IsObject, IsVector ], 0,
function(G,M,c)
return FastExtSQ(G, M, c,true );
#was:
#C := CollectorSQ( G, M, false );
#return ExtensionSQ(C,G,M,c);
end);
#############################################################################
##
#M ExtensionNC( G, M, c )
##
InstallMethod( ExtensionNC, "generic method for pc groups", true,
[ CanEasilyComputePcgs, IsObject, IsVector ], 0,
function(G,M,c)
return FastExtSQ(G, M, c,false );
end);
#############################################################################
##
#M Extensions( G, M )
##
InstallMethod( Extensions,
"generic method for pc groups",
true,
[ CanEasilyComputePcgs, IsObject],
0,
function( G, M )
local C, ext, co, cc, c, i;
C := CollectorSQ( G, M, false );
ext := [ ExtensionSQ( C, G, M, 0 ) ];
# compute the two cocycles
co := TwoCohomologySQ( C, G, M );
if Length( co ) = 0 then return
[SplitExtension(G,M)];
fi;
cc := VectorSpace( M.field, co );
for i in [2..Size(cc)] do
c := AsList( cc )[i];
Add( ext, ExtensionSQ( C, G, M, c ) );
od;
return ext;
end );
InstallGlobalFunction(EXPermutationActionPairs,function(D)
local ag, p1iso, agp, p2iso, DP, p1, p2, gens, genimgs, triso,s,i,u,opt,
gp2,pc1,pc2;
if HasDirectProductInfo(D) then
ag:=DirectProductInfo(D).groups[1];
s:=Size(ag);
if not HasNiceMonomorphism(ag) then
# If this is the first time we use it,
# copy group to avoid carrying too much cruft later.
ag:=Group(GeneratorsOfGroup(ag),One(ag));
SetIsGroupOfAutomorphismsFiniteGroup(ag,true);
SetSize(ag,s);
fi;
IsGroupOfAutomorphismsFiniteGroup(ag);
p1iso:=IsomorphismPermGroup(ag);
agp:=Image(p1iso);
# are both groups solvable?
p2iso:=IsomorphismPermGroup(DirectProductInfo(D).groups[2]);
gp2:=ImagesSource(p2iso);
if IsSolvableGroup(gp2) and IsSolvableGroup(agp) then
# both groups are solvable -- go solvable
pc1:=IsomorphismPcGroup(agp);
pc2:=IsomorphismPcGroup(gp2);
DP:=DirectProduct(ImagesSource(pc1),ImagesSource(pc2));
p1:=Projection(DP,1);
p2:=Projection(DP,2);
gens:=Pcgs(DP);
genimgs:=List(gens,
i->ImagesRepresentative(Embedding(D,1),
PreImagesRepresentative(p1iso,
PreImagesRepresentative(pc1,ImagesRepresentative(p1,i))))
*ImagesRepresentative(Embedding(D,2),
PreImagesRepresentative(p2iso,
PreImagesRepresentative(pc2,ImagesRepresentative(p2,i)))) );
else
opt:=rec(limit:=s,random:=1);
if HasBaseOfGroup(agp) then
opt.knownBase:=BaseOfGroup(agp);
fi;
#p1iso:=p1iso*SmallerDegreePermutationRepresentation(agp);
EraseNaturalHomomorphismsPool(agp);
if s>1 then
repeat
u:=Group(());
gens:=[];
for i in GeneratorsOfGroup(agp) do
if Size(u)<s and not i in u then
Add(gens,i);
u:=DoClosurePrmGp(u,[i],opt);
fi;
od;
if HasBaseOfGroup(agp) then
SetBaseOfGroup(u,BaseOfGroup(agp));
fi;
#Print("rep ",Size(u)," ",s,"\n");
until Size(u)=s;
agp:=u;
else
gens:=GeneratorsOfGroup(agp);
fi;
Info( InfoMatOrb, 1, "found ",Length(gens)," generators");
DP:=DirectProduct(agp,gp2);
# SetIsSolvableGroup(DP,IsSolvableGroup(agp)
# and IsSolvableGroup(ImagesSource(p2iso)));
p1:=Projection(DP,1);
p2:=Projection(DP,2);
# if IsSolvableGroup(DP) then
# gens:=Pcgs(DP);
# else
gens:=GeneratorsOfGroup(DP);
# fi;
Unbind(ag);Unbind(agp);
genimgs:=List(gens,
i->ImagesRepresentative(Embedding(D,1),
PreImagesRepresentative(p1iso,ImagesRepresentative(p1,i)))
*ImagesRepresentative(Embedding(D,2),
PreImagesRepresentative(p2iso,ImagesRepresentative(p2,i))) );
fi;
triso:=GroupHomomorphismByImagesNC(DP,D,gens,genimgs);
SetIsBijective(triso,true);
return rec(pairgens:=genimgs,
permgens:=gens,
isomorphism:=triso,
permgroup:=DP);
else
return false;
fi;
end);
InstallGlobalFunction(EXReducePermutationActionPairs,function(r)
local hom, sel, u, gens, i;
if IsPcgs(r.permgens) then
hom:=true; # dummy, nothing to do here
elif IsSolvableGroup(r.permgroup) then
hom:=IsomorphismPcGroup(r.permgroup);
r.permgroup:=Image(hom,r.permgroup);
r.permgens:=List(r.permgens,i->Image(hom,i));
if IsBound(r.isomorphism) then
r.isomorphism:=RestrictedInverseGeneralMapping(hom)*r.isomorphism;
fi;
else
hom:=SmallerDegreePermutationRepresentation(r.permgroup:cheap);
if NrMovedPoints(Image(hom))<NrMovedPoints(r.permgroup) then
r.permgroup:=Image(hom,r.permgroup);
r.permgens:=List(r.permgens,i->Image(hom,i));
if IsBound(r.isomorphism) then
r.isomorphism:=RestrictedInverseGeneralMapping(hom)*r.isomorphism;
fi;
fi;
# try to reduce nr. of generators
sel:=[];
u:=TrivialSubgroup(r.permgroup);
gens:=r.permgens;
for i in Reversed([1..Length(gens)]) do
if not gens[i] in u then
u:=ClosureSubgroupNC(u,gens[i]);
Add(sel,i);
fi;
od;
for i in Reversed(sel) do
if Size(r.permgroup)=Size(Difference(sel,[i])) then
RemoveSet(sel,i);
fi;
od;
if Length(sel)<Length(gens) then
#Print("Reduce nrgens from ",Length(gens)," to ",Length(sel),"\n");
r.permgens:=r.permgens{sel};
r.pairgens:=r.pairgens{sel};
fi;
fi;
end);
############################################################################
##
#F CompatiblePairs( G, M )
#F CompatiblePairs( G, M, D ) ... D <= Aut(G) x GL
#F CompatiblePairs( G, M, D, flag ) ... D <= Aut(G) x GL normalises K
##
InstallGlobalFunction( CompatiblePairs, function( arg )
local G, M, Mgrp, oper, A, B, D, translate, gens, genimgs, triso, K, K1,
K2, f, tmp, Ggens, pcgs, l, idx, u, tup,Dos,elmlist,preimlist,pows,
baspt,newimgs,i,j,basicact,neu;
# catch arguments
G := arg[1];
M := arg[2];
Mgrp := GroupByGenerators( M.generators );
oper := GroupHomomorphismByImagesNC( G, Mgrp, Pcgs(G), M.generators );
# automorphism groups of G and M
if Length( arg ) = 2 then
Info( InfoCompPairs, 1, " CompP: compute aut group");
A := AutomorphismGroup( G );
B := GL( M.dimension, Characteristic( M.field ) );
D := DirectProduct( A, B );
else
D := arg[3];
A := DirectProductInfo(D).groups[1];
fi;
# the trivial case
if IsBound( M.isCentral ) and M.isCentral then
return D;
fi;
# do we translate D in a permutation group?
translate:=EXPermutationActionPairs(D);
if translate<>false then
D:=translate.permgroup;
gens:=translate.permgens;
genimgs:=translate.pairgens;
triso:=translate.isomorphism;
translate:=true;
else
gens:=GeneratorsOfGroup(D);
genimgs:=gens;
fi;
Dos:=Size(D);
# compute stabilizer of K in A
if Length( arg ) <= 3 or not arg[4] then
# get kernel of oper
K := KernelOfMultiplicativeGeneralMapping( oper );
if Size(K)>1 then
# get its stabilizer
if IsPcGroup(K) then
K1:=CanonicalPcgsWrtFamilyPcgs(Centre(K));
K2:=CanonicalPcgsWrtFamilyPcgs(K);
f := function( pt, a )
return CanonicalPcgsWrtFamilyPcgs(Group(List(pt,i->Image( a[1], i ))));
end;
else
K1:=Centre(K);
K2:=K;
f := function( pt, a ) return Image( a[1], pt ); end;
fi;
if Length(K1)>0 and K1<>K2 then
tmp := Stabilizer( D, K1,gens,genimgs, f );
if Size(tmp)<Size(D) then
Info( InfoMatOrb, 1, " CompP: found orbit of centre of length ",
Size(D)/Size( tmp ));
D := tmp;
if translate<>false then
if HasIsSolvableGroup(D) and IsSolvableGroup(D) then
gens:=Pcgs(D);
else
gens:=GeneratorsOfGroup(D);
fi;
genimgs:=List(gens,i->ImageElm(triso,i));
translate:=rec(pairgens:=genimgs,
permgens:=gens,
isomorphism:=triso,
permgroup:=D);
EXReducePermutationActionPairs(translate);
gens:=translate.permgens;
genimgs:=translate.pairgens;
triso:=translate.isomorphism;
D:=translate.permgroup;
else
gens:=GeneratorsOfGroup(D);
genimgs:=gens;
fi;
fi;
tmp:=false; # clear memory
fi;
tmp := Stabilizer( D, K2,gens,genimgs, f );
if Size(tmp)<Size(D) then
Info( InfoMatOrb, 1, " CompP: found orbit of length ",
Size(D)/Size(tmp));
D := tmp;
if translate<>false then
if HasIsSolvableGroup(D) and IsSolvableGroup(D) then
gens:=Pcgs(D);
else
gens:=GeneratorsOfGroup(D);
fi;
genimgs:=List(gens,i->ImageElm(triso,i));
translate:=rec(pairgens:=genimgs,
permgens:=gens,
isomorphism:=triso,
permgroup:=D);
EXReducePermutationActionPairs(translate);
gens:=translate.permgens;
genimgs:=translate.pairgens;
triso:=translate.isomorphism;
D:=translate.permgroup;
else
gens:=GeneratorsOfGroup(D);
genimgs:=gens;
fi;
fi;
tmp:=false; # clear memory
fi;
fi;
# compute stabilizer of M.generators in D
Ggens:=Pcgs(G);
basicact:=function( tup, elm )
local gens;
#gens := List( tup[1], x -> PreImagesRepresentative( elm[1], x ) );
#gens := List( gens, x -> MappedPcElement( x, tup[1], tup[2] ) );
gens := List( Ggens, x -> PreImagesRepresentative( elm[1], x ) );
gens := List( gens, x -> MappedPcElement( x, Ggens, tup ) );
gens := List( gens, x -> x ^ elm[2] );
return gens;
#return Tuple( [tup[1], gens] );
end;
if Size(G)>20000 then
# if G is too large we cannot write out elements
elmlist:=fail;
f:=basicact;
else
elmlist:=[];
tmp:=List(genimgs,x->x[1]);
preimlist:=List(tmp,x->[x,List(Ggens,y->PreImagesRepresentative(x,y))]);
f:=function( tup, elm )
local gens,i,p;
p:=PositionProperty(preimlist,x->IsIdenticalObj(x[1],elm[1]));
if p=fail then
gens := List( Ggens, x -> PreImagesRepresentative( elm[1], x ) );
else
gens:=preimlist[p][2];
fi;
gens:=List(gens,x->TracedPointPcElement(x,Ggens,elmlist{tup},baspt));
gens:=List(gens,x->x^elm[2]);
return gens;
# tup:=ShallowCopy(tup); # get memory
# avoid duplicate matrices
# for i in [1..Length(gens)] do
# p:=PositionSorted(elmlist,gens[i]);
# if p<>fail and p<=Length(elmlist) and elmlist[p]=gens[i] then
# tup[i]:=p;
# else
# AddSet(elmlist,gens[i]);
# p:=PositionSorted(elmlist,gens[i]);
# tup[i]:=p;
# fi;
# od;
# return tup;
end;
fi;
# build tails of the pcgs that are closed under automorphisms
pcgs:=Pcgs(G);
l:=Length(Pcgs(G))+1;
repeat
Unbind(tmp);
repeat
l:=l-1;
idx:=[l..Length(pcgs)];
u:=SubgroupNC(G,pcgs{idx});
until ForAll(GeneratorsOfGroup(u),
i->ForAll(GeneratorsOfGroup(A),j->Image(j,i) in u));
Ggens:=InducedPcgsByPcSequence(pcgs,pcgs{idx});
tup:=M.generators{idx};
if elmlist<>fail then
tmp:=List(genimgs,x->x[1]);
preimlist:=List(tmp,x->[x,List(Ggens,y->PreImagesRepresentative(x,y))]);
# ensure we also account for action
u:=Group(tup);
elmlist:=AsSSortedList(u);
tmp:=SmallGeneratingSet(u);
i:=1;
while elmlist<>fail and i<=Length(tmp) do
j:=1;
while j<=Length(genimgs) do
neu:=tmp[i]^genimgs[j][2];
if elmlist<>fail and not neu in elmlist then
u:=ClosureGroup(u,neu);
if Size(u)>50000 then
# catch cases of too many elements.
elmlist:=fail;
f:=basicact;
j:=Length(genimgs)+1;
else
elmlist:=AsSSortedList(u);
if Length(SmallGeneratingSet(u))<Length(tmp) then
tmp:=SmallGeneratingSet(u);
i:=0;
j:=Length(genimgs)+1; # force loop reset
else
tmp:=Concatenation(tmp,[neu]);
fi;
fi;
fi;
j:=j+1;
od;
i:=i+1;
od;
if elmlist<>fail then
baspt:=Position(elmlist,One(u));
# describe how second part acts on matrices by conjugation
newimgs:=List(genimgs,
x->DirectProductElement([x[1],Permutation(x[2],elmlist,OnPoints)]));
Assert(1,ForAll(newimgs,x->x[2]<>fail));
tup:=List(tup,x->Position(elmlist,x));
elmlist:=List(elmlist,x->Permutation(x,elmlist,OnRight));
pows:=NextPrimeInt(Length(elmlist)-20); # we are likely sparse, so
# not being perfect is not likely to do a hash conflict
pows:=List([0..Length(tup)],x->pows^x);
tmp:=[D, rec(hashfun:= lst->lst*pows),tup, gens,newimgs, f ];
# use `op' to get in the fake domain with the hashfun
tmp := StabilizerOp( D, rec(hashfun:= lst->lst*pows),tup,
gens,newimgs, f );
else
tmp := Stabilizer( D, tup,gens,genimgs, f );
fi;
else
tmp := Stabilizer( D, tup,gens,genimgs, f );
fi;
Info( InfoMatOrb, 1, " CompP: ",l,"-tail found orbit of length ",
Size(D)/Size(tmp));
if Size(tmp)<Size(D) then
D:=tmp;
if IsPcgs(gens) then
gens:=InducedPcgs(gens,tmp);
else
gens:=SmallGeneratingSet(tmp);
fi;
genimgs:=List(gens,i->ImageElm(triso,i));
if translate<>false then
translate:=rec(pairgens:=genimgs,
permgens:=gens,
isomorphism:=triso,
permgroup:=D);
EXReducePermutationActionPairs(translate);
gens:=translate.permgens;
genimgs:=translate.pairgens;
triso:=translate.isomorphism;
D:=translate.permgroup;
fi;
fi;
until l=1;
if translate<>false then
l:=Size(D);
if Length(gens)>3 then
# reduce generator number
u:=SmallGeneratingSet(D);
if IsSubset(gens,u) then
Info( InfoMatOrb, 3, "Reduce generators subset");
idx:=List(u,x->Position(gens,x));
gens:=gens{idx};
genimgs:=genimgs{idx};
else
Info( InfoMatOrb, 3, "Reduce generators new words");
gens:=u;
genimgs:=List(gens,i->ImageElm(triso,i));
fi;
fi;
tmp:=SubgroupNC(Range(triso),genimgs);
SetIsGroupOfAutomorphismsFiniteGroup(tmp,true);
SetSize(tmp,l);
# cache the faithful permutation representation in case we need it
# later
tmp!.permrep:=rec(pairgens:=genimgs,
permgens:=gens,
permgroup:=D);
D:=tmp;
fi;
Info( InfoMatOrb, 1, "Total index: ",Dos/Size(D));
return D;
end );
#############################################################################
##
#F MatrixOperationOfCPGroup( cc, gens )
##
MatrixOperationOfCPGroup := function( cc, gens )
local mats, base, pcgs, ords, imgs, n, d, fpgens, fprels, H, pcgsH,
l, g, imgl, k, i, j, rel, tail, m, tails, prei, h,field;
mats := List( gens, x -> [] );
base := Basis( Image( cc.cohom ) );
prei := List( base, x -> PreImagesRepresentative( cc.cohom, x ) );
pcgs := Pcgs( cc.group );
ords := RelativeOrders( pcgs );
imgs := List( gens, x -> List( pcgs, y -> y^Inverse( x[1] ) ) );
n := Length( pcgs );
d := cc.module.dimension;
field:=cc.module.field;
fpgens := GeneratorsOfGroup( cc.presentation.group );
fprels := cc.presentation.relators;
# loop over base elements and compute images under operation
for h in [1..Length(base)] do
H := ExtensionSQ( cc.collector, cc.group, cc.module, prei[h] );
pcgsH := Pcgs( H );
# loop over generators
for l in [1..Length(gens)] do
g := gens[l];
imgl := List( imgs[l], x -> MappedPcElement( x, pcgs, pcgsH ) );
if imgl <> pcgs then
# compute tails of relators in H
k := 0;
tails := [];
for i in [1..Length(pcgs)] do
for j in [1..i] do
# compute tail of relator
k := k + 1;
rel := fprels[k];
tail := MappedWord( rel, fpgens, imgl );
# conjugating element
if not IsBound( cc.module.isCentral ) or
not cc.module.isCentral then
if i = j then
m := imgl[i]^ords[i];
else
m := imgl[i]^imgl[j];
fi;
tail := tail^m;
fi;
tail := ExponentsOfPcElement(pcgsH,tail,[n+1..n+d]);
tail := tail * g[2];
# convert tail to compressed format ...
if IsHPCGAP then
if Size(field)<=256 then
tail := CopyToVectorRepNC(tail,Size(field));
fi;
else
ConvertToVectorRepNC(tail,field);
fi;
# ... and append tail to tails; we have to
# treat the case that tails is still empty separately,
# because right now, GAP does not support empty
# compressed vectors; hence tails is an empty plist,
# and Append will leave it at that.
if Length(tails) = 0 then
tails := tail;
else
Append( tails, tail );
fi;
od;
od;
else
Error("not yet done");
fi;
tails := Image( cc.cohom, tails );
Add( mats[l], tails );
od;
od;
return List(mats,i->ImmutableMatrix(field,i));
end;
#############################################################################
##
#M ExtensionRepresentatives( G, M, C )
##
InstallMethod( ExtensionRepresentatives,
"generic method for pc groups",
true,
[ CanEasilyComputePcgs, IsRecord, IsGroup ],
0,
function( G, M, C )
local cc, ext, mats, Mgrp, orbs, c;
cc := TwoCohomology( G, M );
# catch the trivial case
if Dimension(Image(cc.cohom)) = 0 then
return [ExtensionSQ( cc.collector, G, M, 0 )];
elif Dimension( Image(cc.cohom)) = 1 then
c := Basis(Image(cc.cohom))[1];
c := PreImagesRepresentative(cc.cohom, c);
return [ExtensionSQ( cc.collector, G, M, 0 ),
ExtensionSQ( cc.collector, G, M, c )];
fi;
mats := MatrixOperationOfCPGroup( cc, GeneratorsOfGroup( C ) );
# compute orbit of mats on H^2( G, M )
Mgrp := GroupByGenerators( mats );
orbs := OrbitsDomain( Mgrp, Image(cc.cohom), OnRight );
orbs := List( orbs, x -> PreImagesRepresentative( cc.cohom, x[1] ) );
ext := List( orbs, x -> ExtensionSQ( cc.collector, G, M, x ) );
return ext;
end);
#############################################################################
##
#F MyIntCoefficients( p, d, w )
##
MyIntCoefficients := function( p, d, w )
local v, int, i;
v := IntVecFFE( w );
int := 0;
for i in [1..d] do
int := int * p + v[i];
od;
return int;
end;
#############################################################################
##
#F MatOrbs( mats, dim, field )
##
MatOrbs := function( mats, dim, field )
local p, q, r, l, n, seen, reps, rest, i, v, orb, j, w, im, h, mat, rep;
# set up
p := Characteristic( field );
q := p^dim;
r := p^dim - 1;
l := List( [1..dim], x -> p );
n := Length( mats );
# set up large boolean list
seen := [];
seen[q] := false;
for i in [1..q-1] do seen[i] := false; od;
IsBlist( seen );
reps := [];
rest := r;
for i in [1..r] do
if not seen[i] then
seen[i] := true;
v := CoefficientsMultiadic( l, i );
orb := [v];
rest := rest - 1;
j := 1;
rep := v;
Add( reps, rep );
while j <= Length( orb ) do
w := orb[j];
for mat in mats do
im := w * mat;
h := MyIntCoefficients( p, dim, im );
if not seen[h] then
seen[h] := true;
rest := rest - 1;
Add( orb, im );
fi;
od;
if rest = 0 then j := Length( orb ); fi;
j := j + 1;
od;
Info( InfoExtReps, 3, "found orbit of length: ", Length(orb),
" remaining points: ",rest);
fi;
od;
return reps * One( field );
end;
#############################################################################
##
#F NonSplitExtensions( G, M [, reduce] )
##
NonSplitExtensions := function( arg )
local G, M, C, cc, cohom, mats, CP, all, red, c;
# catch arguments
G := arg[1];
M := arg[2];
# compute H^2(G, M)
cc := TwoCohomology( G, M );
C := cc.collector;
Info( InfoExtReps, 1, " dim(M) = ",M.dimension,
" char(M) = ", Characteristic(M.field),
" dim(H2) = ", Dimension(Image(cc.cohom)));
# catch the trivial cases
if Dimension( Image( cc.cohom ) ) = 0 then
all := [];
red := true;
elif Dimension( Image(cc.cohom ) ) = 1 then
c := PreImagesRepresentative(cc.cohom, Basis(Image(cc.cohom))[1]);
all := [ExtensionSQ( C, G, M, c)];
red := true;
# if reduction is suppressed
elif IsBound( arg[3] ) and not arg[3] then
all := NormedRowVectors( Image(cc.cohom) );
all := List( all, x -> ExtensionSQ(cohom.collector, G, M,
PreImagesRepresentative(cc.cohom,x )));
red := false;
# sometimes we do not want to reduce
elif not IsBound( arg[3] )
and Size(Image(cc.cohom)) < 10
and not (HasIsFrattiniFree( G ) and IsFrattiniFree( G ))
and not HasAutomorphismGroup( G )
then
all := NormedRowVectors( Image(cc.cohom) );
all := List( all, x -> ExtensionSQ(cc.collector, G, M,
PreImagesRepresentative(cc.cohom, x )));
red := false;
# then we want to reduce
else
Info( InfoExtReps, 2, " Ext: compute compatible pairs");
CP := CompatiblePairs( G, M );
Info( InfoExtReps, 2, " Ext: compute linear action");
mats := MatrixOperationOfCPGroup( cc, GeneratorsOfGroup( CP ) );
Info( InfoExtReps, 2, " Ext: compute orbits ");
all := MatOrbs( mats, Length(mats[1]) , M.field );
red := true;
Info( InfoExtReps, 2, " Ext: found ",Length(all)," orbits ");
# create extensions and add info
all := List( all, x -> ExtensionSQ(cc.collector, G, M,
PreImagesRepresentative(cc.cohom, x )));
fi;
if red then
Info( InfoExtReps, 1, " found ",Length(all),
" extensions - reduced");
else
Info( InfoExtReps, 1, " found ",Length(all)," extensions ");
fi;
return rec( groups := all, reduced := red );
end;
#############################################################################
##
#F SplitExtension( G, M )
#F SplitExtension( G, aut, N )
##
InstallMethod( SplitExtension,
"generic method for pc groups",
true,
[ CanEasilyComputePcgs, IsObject ],
0,
function( G, M )
return Extension( G, M, 0 );
end );
InstallOtherMethod( SplitExtension,
"generic method for pc groups",
true,
[ CanEasilyComputePcgs, IsObject, CanEasilyComputePcgs ],
0,
function( G, aut, N )
local pcgsG, fpg, n, gensG, pcgsN, fpn, d, gensN, F, gensF, relators,
rel, new, g, e, t, l, i, j, k, H, m, relsN, relsG;
pcgsG := Pcgs( G );
fpg := Range( IsomorphismFpGroupByPcgs( pcgsG, "g" ) );
n := Length( pcgsG );
gensG := GeneratorsOfGroup( FreeGroupOfFpGroup( fpg ) );
relsG := RelatorsOfFpGroup( fpg );
pcgsN := Pcgs( N );
fpn := Range( IsomorphismFpGroupByPcgs( pcgsN, "n" ) );
d := Length( pcgsN );
gensN := GeneratorsOfGroup( FreeGroupOfFpGroup( fpn ) );
relsN := RelatorsOfFpGroup( fpn );
F := FreeGroup(IsSyllableWordsFamily, n + d );
gensF := GeneratorsOfGroup( F );
relators := [];
# relators of G
for rel in relsG do
new := MappedWord( rel, gensG, gensF{[1..n]} );
Add( relators, new );
od;
# operation of G on N
for i in [1..n] do
for j in [1..d] do
# left hand side
l := Comm( gensF[n+j], gensF[i] );
# right hand side
g := Image( aut, pcgsG[i] );
m := Image( g, pcgsN[j] );
e := ExponentsOfPcElement( pcgsN, (pcgsN[j]^-1 * m)^-1 );
t := One( F );
for k in [1..d] do
t := t * gensF[n+k]^e[k];
od;
# add new relator
Add( relators, l * t );
od;
od;
# relators of N
for rel in relsN do
new := MappedWord( rel, gensN, gensF{[n+1..n+d]} );
Add( relators, new );
od;
H := PcGroupFpGroup( F / relators );
SetModuleOfExtension( H, Subgroup(H, Pcgs(H){[n+1..n+d]} ) );
return H;
end);