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#############################################################################
##
#W ghomfp.gi GAP library Alexander Hulpke
##
#Y (C) 2000 School Math and Comp. Sci., University of St Andrews, Scotland
#Y Copyright (C) 2002 The GAP Group
##
#############################################################################
##
## methods for homomorphisms that map the standard generators -- no
## rewriting necessary
#############################################################################
##
#M ImagesRepresentative( <hom>, <elm> )
##
InstallMethod( ImagesRepresentative,
"map from fp group or free group, use 'MappedWord'",
FamSourceEqFamElm, [ IsFromFpGroupStdGensGeneralMappingByImages,
IsMultiplicativeElementWithInverse ], 0,
function( hom, elm )
local mapi;
mapi:=MappingGeneratorsImages(hom);
return MappedWord(elm,mapi[1],mapi[2]);
end);
#############################################################################
##
#M IsSingleValued
##
InstallMethod( IsSingleValued,
"map from fp group or free group on arbitrary gens: rewrite",
true,
[IsFromFpGroupGeneralMappingByImages and HasMappingGeneratorsImages],0,
function(hom)
local m, fp, s, sg, o, gi;
m:=MappingGeneratorsImages(hom);
fp:=IsomorphismFpGroupByGenerators(Source(hom),m[1]);
s:=Image(fp);
sg:=FreeGeneratorsOfFpGroup(s);
o:=One(Range(hom));
gi:=m[2];
return ForAll(RelatorsOfFpGroup(s),i->MappedWord(i,sg,gi)=o);
end);
InstallMethod( IsSingleValued,
"map from whole fp group or free group, given on std. gens: test relators",
true,
[IsFromFpGroupStdGensGeneralMappingByImages],0,
function(hom)
local s,sg,o,gi;
s:=Source(hom);
if not IsWholeFamily(s) then
TryNextMethod();
fi;
if IsFreeGroup(s) then
return true;
fi;
sg:=FreeGeneratorsOfFpGroup(s);
o:=One(Range(hom));
# take the images corresponding to the free gens in case of reordering or
# duplicates
gi:=MappingGeneratorsImages(hom)[2]{hom!.genpositions};
return ForAll(RelatorsOfFpGroup(s),i->MappedWord(i,sg,gi)=o);
end);
InstallMethod( IsSingleValued,
"map from whole fp group or free group to perm, std. gens: test relators",
true,
[IsFromFpGroupStdGensGeneralMappingByImages and
IsToPermGroupGeneralMappingByImages],0,
function(hom)
local s, bas, sg, o, gi, l, p, rel, start, i;
s:=Source(hom);
if not IsWholeFamily(s) then
TryNextMethod();
fi;
if IsFreeGroup(s) then
return true;
fi;
bas:=BaseStabChain(StabChainMutable(Range(hom)));
sg:=FreeGeneratorsOfFpGroup(s);
o:=One(Range(hom));
# take the images corresponding to the free gens in case of reordering or
# duplicates
gi:=MappingGeneratorsImages(hom)[2]{hom!.genpositions};
for rel in RelatorsOfFpGroup(s) do
l:=LetterRepAssocWord(rel);
for start in bas do
p:=start;
for i in l do
if i>0 then
p:=p^gi[i];
else
p:=p/gi[-i];
fi;
od;
if p<>start then
return false;
fi;
od;
od;
return true;
end);
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <hom> )
##
InstallMethod( KernelOfMultiplicativeGeneralMapping,
"from fp/free group, std. gens., to perm group",
true, [ IsFromFpGroupGeneralMapping
and IsToPermGroupGeneralMappingByImages ],0,
function(hom)
local f,p,t,orbs,o,cor,u,frg;
f:=Source(hom);
if not (HasIsWholeFamily(f) and IsWholeFamily(f)) then
TryNextMethod();
fi;
frg:=FreeGeneratorsOfFpGroup(f);
t:=List(GeneratorsOfGroup(f),i->Image(hom,i));
p:=SubgroupNC(Range(hom),t);
Assert(1,GeneratorsOfGroup(p)=t);
# construct coset table
t:=[];
orbs:=OrbitsDomain(p,MovedPoints(p));
cor:=f;
for o in orbs do
u:=SubgroupOfWholeGroupByQuotientSubgroup(FamilyObj(f),
p,Core(p,Stabilizer(p,o[1])));
cor:=Intersection(cor,u);
od;
if IsIdenticalObj(cor,f) then # in case we get a wrong parent
SetIsNormalInParent(cor,true);
fi;
return cor;
end);
#############################################################################
##
## methods for arbitrary mappings. We must use rewriting.
##
#############################################################################
##
#F SecondaryImagesAugmentedCosetTable(<aug>,<gens>,<genimages>)
##
InstallGlobalFunction(SecondaryImagesAugmentedCosetTable,function(aug)
local si,sw,i,ug,tt,p;
if not IsBound(aug.secondaryImages) then
# set the secondary generators images
si:=[];
ug:=List(aug.homgens,UnderlyingElement);
tt:=GeneratorTranslationAugmentedCosetTable(aug);
sw:=SecondaryGeneratorWordsAugmentedCosetTable(aug);
for i in [1..Length(tt)] do
# get the word representative for the secondary generator
p:=Position(ug,UnderlyingElement(sw[i]));
if p<>fail then
Add(si,aug.homgenims[p]);
else
# its not. We must map the image from the primary generators images.
# For this we use that their images must be given already in `si', as
# the primary generators come first.
Add(si,MappedWord(tt[i],aug.primarySubgroupGenerators,
si{[1..Length(aug.primarySubgroupGenerators)]}));
fi;
od;
aug.secondaryImages:=si;
fi;
return aug.secondaryImages;
end);
# test whether evaluating all the secondary images might be sensible.
InstallGlobalFunction(TrySecondaryImages,function(aug)
local p;
p:=aug.primaryImages;
if Length(p)>0 and (
# would it cost too much storage, to store all secondary generators?
(IsPerm(p[1]) and ForAll(p,i->LargestMovedPoint(i)<50)) or
(IsNBitsPcWordRep(p[1])) ) then
aug.secondaryImages:=ShallowCopy(p);
fi;
end);
#############################################################################
##
#M CosetTableFpHom(<hom>)
##
InstallMethod(CosetTableFpHom,"for fp homomorphisms",true,
[ IsFromFpGroupGeneralMappingByImages and IsGroupGeneralMappingByImages],0,
function(hom)
local u,aug,hgu,mapi,w;
# source group with suitable generators
u:=Source(hom);
aug:=false;
mapi:=MappingGeneratorsImages(hom);
hgu:=List(mapi[1],UnderlyingElement);
# construct augmented coset table
w:=FamilyObj(mapi[1])!.wholeGroup;
aug:=NEWTC_CosetEnumerator(FreeGeneratorsOfFpGroup(w),
RelatorsOfFpGroup(w),
hgu,
true);
aug.homgens:=mapi[1];
aug.homgenims:=mapi[2];
# assign the primary generator images
aug.primaryImages:=List(aug.subgens,
i->aug.homgenims[Position(hgu,i)]);
# TODO: possibly re-use an existing augmented table stored already
TrySecondaryImages(aug);
return aug;
end);
#############################################################################
##
#M ImagesRepresentative( <hom>, <elm> )
##
InstallMethod( ImagesRepresentative, "map from (sub)fp group, rewrite",
FamSourceEqFamElm,
[ IsFromFpGroupGeneralMappingByImages and IsGroupGeneralMappingByImages,
IsMultiplicativeElementWithInverse ], 0,
function( hom, word )
local aug,si,r,i,j,tt,ct,cft,c,f,g,ind,e,eval;
# get a coset table
aug:=CosetTableFpHom(hom);
r:=One(Range(hom));
if IsBound(aug.secondaryImages) then
si:=aug.secondaryImages;
elif IsBound(aug.primaryImages) then
si:=aug.primaryImages;
else
Error("no decoding possible");
fi;
word:=UnderlyingElement(word);
if IsBound(aug.isNewAugmentedTable) then
eval:=function(i)
local w,j;
w:=One(si[1]);
if not IsBound(si[i]) then
for j in aug.secondary[i] do
if j<0 then
w:=w/eval(-j);
else
w:=w*eval(j);
fi;
od;
si[i]:=w;
fi;
return si[i];
end;
for i in NEWTC_Rewrite(aug,1,LetterRepAssocWord(word)) do
if i<0 then r:=r/eval(-i);
else r:=r*eval(i);
fi;
od;
return r;
fi;
# old version
# instead of calling `RewriteWord', we rewrite locally in the images.
# this ought to be a bit faster and better on memory.
ct := aug.cosetTable;
cft := aug.cosetFactorTable;
# translation table for group generators to numbers
if not IsBound(aug.transtab) then
# should do better, also cope with inverses
aug.transtab:=List(aug.groupGenerators,i->GeneratorSyllable(i,1));
fi;
tt:=aug.transtab;
c:=1; # current coset
if not IsLetterAssocWordRep(word) then
# syllable version
for i in [1..NrSyllables(word)] do
g:=GeneratorSyllable(word,i);
e:=ExponentSyllable(word,i);
if e<0 then
ind:=2*aug.transtab[g];
e:=-e;
else
ind:=2*aug.transtab[g]-1;
fi;
for j in [1..e] do
# apply the generator, collect cofactor
f:=cft[ind][c]; # cofactor
if f>0 then
r:=r*DecodedTreeEntry(aug.tree,si,f);
elif f<0 then
r:=r/DecodedTreeEntry(aug.tree,si,-f);
fi;
c:=ct[ind][c]; # new coset number
od;
od;
else
# letter version
word:=LetterRepAssocWord(word);
for i in [1..Length(word)] do
g:=word[i];
if g<0 then
g:=-g;
ind:=2*aug.transtab[g];
else
ind:=2*aug.transtab[g]-1;
fi;
# apply the generator, collect cofactor
f:=cft[ind][c]; # cofactor
if f>0 then
r:=r*DecodedTreeEntry(aug.tree,si,f);
elif f<0 then
r:=r/DecodedTreeEntry(aug.tree,si,-f);
fi;
c:=ct[ind][c]; # new coset number
od;
fi;
# make sure we got back to start
if c<>1 then
Error("<elm> is not contained in the source group");
fi;
return r;
end);
InstallMethod( ImagesRepresentative,
"simple tests on equal words to check whether the `generators' are mapped",
FamSourceEqFamElm,
[ IsFromFpGroupGeneralMappingByImages and IsGroupGeneralMappingByImages,
IsMultiplicativeElementWithInverse ],
# this is a better method than the previous, as it will probably avoid
# rewriting.
1,
function( hom, elm )
local he,ue,p,mapi;
ue:=UnderlyingElement(elm);
if IsLetterAssocWordRep(ue) and IsOne(ue) then
return One(Range(hom));
fi;
mapi:=MappingGeneratorsImages(hom);
p:=PositionProperty(mapi[1],i->IsIdenticalObj(UnderlyingElement(i),ue));
if p<>fail then
return mapi[2][p];
fi;
ue:=ue^-1;
p:=PositionProperty(mapi[1],i->IsIdenticalObj(UnderlyingElement(i),ue));
if p<>fail then
return mapi[2][p]^-1;
fi;
TryNextMethod();
end);
#############################################################################
##
#M KernelOfMultiplicativeGeneralMapping( <hom> )
##
InstallMethod( KernelOfMultiplicativeGeneralMapping, "hom from fp grp", true,
[ IsFromFpGroupGeneralMapping and IsGroupGeneralMapping], 0,
function(hom)
local k;
k:=PreImage(hom,TrivialSubgroup(Range(hom)));
if HasIsSurjective(hom) and IsSurjective( hom ) and
HasIndexInWholeGroup( Source(hom) )
and HasRange(hom) # surjective action homomorphisms do not store
# the range by default
and HasSize( Range( hom ) ) then
SetIndexInWholeGroup( k,
IndexInWholeGroup( Source(hom) ) * Size( Range(hom) ));
fi;
return k;
end);
#############################################################################
##
#M CoKernelOfMultiplicativeGeneralMapping( <hom> )
##
InstallMethod( CoKernelOfMultiplicativeGeneralMapping, "GHBI from fp grp", true,
[ IsFromFpGroupGeneralMappingByImages
and IsGroupGeneralMappingByImages ], 0,
function(map)
local so,fp,isofp,rels,mapi;
# the mapping is on the std. generators. So we just have to evaluate the
# relators in the generators on the genimages and take the normal closure.
so:=Source(map);
mapi:=MappingGeneratorsImages(map);
if Length(GeneratorsOfGroup(so))=0
or ForAll(GeneratorsOfGroup(so),x->IsOne(UnderlyingElement(x))) then
rels:=ShallowCopy(mapi[2]);
else
isofp:=IsomorphismFpGroupByGeneratorsNC(so,mapi[1],"F");
fp:=Range(isofp);
rels:=RelatorsOfFpGroup(fp);
rels:=List(rels,i->MappedWord(i,FreeGeneratorsOfFpGroup(fp),mapi[2]));
fi;
return NormalClosure(Range(map),SubgroupNC(Range(map),rels));
end);
BindGlobal("WreathElm",function(b,l,m)
local n,ran,r,d,p,i,j;
n:=Length(l);
ran:=[1..b];
r:=0;
d:=[];
p:=[];
# base bit
for i in [1..n] do
for j in ran do
p[r+j]:=r+j^l[i];
od;
Add(d,ran+r);
r:=r+b;
od;
# permuter bit
p:=PermList(p)/PermList(Concatenation(Permuted(d,m)));
return p;
end);
InstallGlobalFunction(KuKGenerators,
function(G,beta,alpha)
local q,r,tg,dtg,pemb,ugens,g,gi,d,o,gens,genims,i,gr,img,l,mapi;
q:=Range(beta);
d:=NrMovedPoints(q);
# transversal (sorted)
#better: orbit algo
#r:=ShallowCopy(RightTransversal(q,qu));
#Sort(r,function(a,b) return 1^a<1^b;end);
#r:=List(r,i->PreImagesRepresentative(beta,i));
# compute transversal with short words from orbit algorithm on points
o:=[1];
mapi:=MappingGeneratorsImages(beta);
gens:=mapi[1];
genims:=mapi[2];
gr:=[1..Length(gens)];
r:=[One(gens[1])];
i:=1;
while i<=Length(o) do
for g in gr do
img:=o[i]^genims[g];
if not img in o then
Add(o,img);
Add(r,r[i]*gens[g]);
fi;
od;
i:=i+1;
od;
SortParallel(o,r); # indices in right position -- this *is* important
# because we use the index to get the transversal representative!
tg:=Range(alpha);
if IsPermGroup(tg) then
pemb:=IdentityMapping(tg);
dtg:=LargestMovedPoint(Range(pemb));
elif Size(tg)<20 then
pemb:=IsomorphismPermGroup(tg);
dtg:=LargestMovedPoint(Range(pemb));
else
pemb:=IdentityMapping(tg);
dtg:=-1;
fi;
if dtg=0 then
dtg:=1; # the darn trivial group again.
fi;
# images of the generators in the wreath
ugens:=[];
for g in GeneratorsOfGroup(G) do
gi:=ImagesRepresentative(beta,g);
l:=[];
for i in [1..d] do
l[i]:=ImagesRepresentative(pemb,
ImagesRepresentative(alpha,r[i]*g/r[i^gi]));
od;
Add(ugens,WreathElm(dtg,l,gi) );
od;
return ugens;
end);
#############################################################################
##
#M InducedRepFpGroup(<hom>,<u> )
##
## induce <hom> def. on <u> up to the full group
BindGlobal("InducedRepFpGroup",function(thom,s)
local t,w,c,q,chom,tg,hi,u;
w:=FamilyObj(s)!.wholeGroup;
# permutation action on the cosets
c:=CosetTableInWholeGroup(s);
c:=List(c{[1,3..Length(c)-1]},PermList);
q:=Group(c,()); # `c' arose from `PermList'
chom:=GroupHomomorphismByImagesNC(w,q,GeneratorsOfGroup(w),c);
if Size(q)=1 then
# degenerate case
return thom;
else
u:=KuKGenerators(w,chom,thom);
fi;
q:=GroupWithGenerators(u,()); # `u' arose from `KuKGenerators'
return GroupHomomorphismByImagesNC(w,q,GeneratorsOfGroup(w),u);
end);
BindGlobal("IsTransPermStab1",function(G,U)
return IsPermGroup(G) and IsTransitive(G,MovedPoints(G))
and (1 in MovedPoints(G)) and Length(Orbit(U,1))=1
and Size(G)/Size(U)=Length(MovedPoints(G));
end);
#############################################################################
##
#M PreImagesSet( <hom>, <u> )
##
InstallMethod( PreImagesSet, "map from (sub)group of fp group",
CollFamRangeEqFamElms,
[ IsFromFpGroupHomomorphism,IsGroup ],0,
function(hom,u)
local s,t,p,w,c,q,chom,tg,thom,hi,i,lp,max;
s:=Source(hom);
if HasIsWholeFamily(s) and IsWholeFamily(s) then
t:=List(GeneratorsOfGroup(s),i->Image(hom,i));
if IsPermGroup(Range(hom)) and LargestMovedPoint(t)<>NrMovedPoints(t) then
c:=MappingPermListList(MovedPoints(t),[1..NrMovedPoints(t)]);
t:=List(t,i->i^c);
u:=u^c;
else
c:=false;
fi;
p:=GroupWithGenerators(t);
if HasImagesSource(hom) and HasSize(Image(hom)) then
SetSize(p,Size(Image(hom)));
fi;
if c=false then
SetParent(p,Range(hom));
fi;
if HasIsSurjective(hom) and IsSurjective(hom) then
SetIndexInParent(p,1);
fi;
return SubgroupOfWholeGroupByQuotientSubgroup(FamilyObj(s),p,u);
fi;
w:=FamilyObj(s)!.wholeGroup;
# permutation action on the cosets
if IsBound(s!.quot) and IsTransPermStab1(s!.quot,s!.sub) then
q:=s!.quot;
c:=GeneratorsOfGroup(q);
else
c:=CosetTableInWholeGroup(s);
c:=List(c{[1,3..Length(c)-1]},PermList);
q:=Group(c,()); # `c' arose from `PermList'
if IsBound(s!.quot) and HasSize(s!.quot) then
# transfer size information
StabChainOp(q,rec(limit:=Size(s!.quot)));
fi;
fi;
chom:=GroupHomomorphismByImagesNC(w,q,GeneratorsOfGroup(w),c);
# action on cosets of U
hi:=Image(hom);
if Index(hi,u)<>infinity then
t:=CosetTableBySubgroup(hi,u);
t:=List(t{[1,3..Length(t)-1]},PermList);
tg:=Group(t,()); # `t' arose from `PermList'
thom:=hom*GroupHomomorphismByImagesNC(hi,tg,GeneratorsOfGroup(hi),t);
# don't use size -- could be expensive
if ForAll(GeneratorsOfGroup(q),IsOne) then
# degenerate case
u:=List(GeneratorsOfGroup(w),i->ImageElm(thom,i));
u:=GroupWithGenerators(u,());
else
u:=KuKGenerators(w,chom,thom);
# could the group be too expensive?
if (not IsBound(s!.quot)) or
(IsPermGroup(s!.quot)
and Size(s!.quot)>10^50 and NrMovedPoints(s!.quot)>10000) then
t:=[];
max:=LargestMovedPoint(u);
for i in u do
#Add(t,ListPerm(i));
lp:=ListPerm(i);
while Length(lp)<max do Add(lp,Length(lp)+1);od;
Add(t,lp);
#Add(t,ListPerm(i^-1));
lp:=ListPerm(i^-1);
while Length(lp)<max do Add(lp,Length(lp)+1);od;
Add(t,lp);
od;
return SubgroupOfWholeGroupByCosetTable(FamilyObj(s),t);
fi;
u:=GroupWithGenerators(u,()); # `u' arose from `KuKGenerators'
# indicate wreath structure
StabChainOp(u,rec(limit:=Size(tg)^NrMovedPoints(q)*Size(q)));
fi;
else
#[hi:u] might be infinite
u:=WreathProduct(hi,q);
Error("infinite");
fi;
return SubgroupOfWholeGroupByQuotientSubgroup(FamilyObj(s),u,Stabilizer(u,1));
end);
#############################################################################
##
#M IsConjugatorIsomorphism( <hom> )
##
InstallMethod( IsConjugatorIsomorphism,
"for a f.p. group general mapping",
true,
[ IsGroupGeneralMapping ], 1,
# There is no filter to test whether source and range of a homomorphism
# are f.p. 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 IsSubgroupFpGroup( s ) then
TryNextMethod();
elif not ( IsGroupHomomorphism( hom ) and IsBijective( hom ) ) then
return false;
elif IsEndoGeneralMapping( hom ) and IsInnerAutomorphism( hom ) then
return true;
fi;
r:= Range( hom );
# Check whether source and range are in the same family.
if FamilyObj( s ) <> FamilyObj( r ) then
return false;
fi;
# Compute a conjugator in the full f.p. group.
G:= FamilyObj( s )!.wholeGroup;
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;
else
return false;
fi;
end );
#############################################################################
##
#M CompositionMapping2( <hom1>, <hom2> ) . . . . . . . . . . . . via images
##
## we override the method for group homomorphisms, to transfer the coset
## table information as well.
InstallMethod( CompositionMapping2,
"for gp. hom. and fp. hom, transferring the coset table",
FamSource1EqFamRange2,
[ IsGroupHomomorphism,
IsGroupHomomorphism and IsFromFpGroupGeneralMappingByImages and
HasCosetTableFpHom], 0,
function( hom1, hom2 )
local map,tab,tab2,i;
if IsNiceMonomorphism(hom2) then
# this is unlikely, but who knows of the things to come...
TryNextMethod();
fi;
if not IsSubset(Source(hom1),ImagesSource(hom2)) then
TryNextMethod();
fi;
map:=MappingGeneratorsImages(hom2);
map:=GroupGeneralMappingByImagesNC( Source( hom2 ), Range( hom1 ),
map[1], List( map[2], img ->
ImagesRepresentative( hom1, img ) ) );
SetIsMapping(map,true);
tab:=CosetTableFpHom(hom2);
tab2:=CopiedAugmentedCosetTable(tab);
tab2.primaryImages:=[];
for i in [1..Length(tab.primaryImages)] do
if IsBound(tab.primaryImages[i]) then
tab2.primaryImages[i]:=ImagesRepresentative(hom1,tab.primaryImages[i]);
fi;
od;
TrySecondaryImages(tab2);
SetCosetTableFpHom(map,tab2);
return map;
end);
#############################################################################
##
## methods for homomorphisms to fp groups.
#############################################################################
##
#M PreImagesRepresentative
##
InstallMethod( PreImagesRepresentative,
"hom. to standard generators of fp group, using 'MappedWord'",
FamRangeEqFamElm,
[IsToFpGroupHomomorphismByImages,IsMultiplicativeElementWithInverse],
# there is no filter indicating the images are standard generators, so we
# must rank higher than the default.
1,
function(hom,elm)
local mapi;
mapi:=MappingGeneratorsImages(hom);
# check, whether we map to the standard generators
if not (HasIsWholeFamily(Range(hom)) and IsWholeFamily(Range(hom)) and
Set(FreeGeneratorsOfFpGroup(Range(hom)))
=Set(List(GeneratorsOfGroup(Range(hom)),UnderlyingElement)) and
IsIdenticalObj(mapi[2],GeneratorsOfGroup(Range(hom))) and
ForAll(List(mapi[2],i->LetterRepAssocWord(UnderlyingElement(i))),
i->Length(i)=1 and i[1]>0) ) then
TryNextMethod();
fi;
if Length(mapi[2])=0 then
mapi:=One(Source(hom));
else
mapi:=MappedWord(elm,mapi[2],mapi[1]);
fi;
return mapi;
end);
#############################################################################
##
## methods to construct homomorphisms to fp groups
##
InstallOtherMethod(IsomorphismFpGroup,"subgroups of fp group",true,
[IsSubgroupFpGroup,IsString],0,
function(u,str)
local aug,w,p,pres,f,fam,opt;
if HasIsWholeFamily(u) and IsWholeFamily(u) then
return IdentityMapping(u);
fi;
# catch trivial case of rank 0 group
if Length(GeneratorsOfGroup(FamilyObj(u)!.wholeGroup))=0 then
return IsomorphismFpGroup(FamilyObj(u)!.wholeGroup,str);
fi;
# get an augmented coset table from the group. Since we don't care about
# any particular generating set, we let the function chose.
aug:=AugmentedCosetTableInWholeGroup(u);
Info( InfoFpGroup, 1, "Presentation with ",
Length(aug.subgroupGenerators), " generators");
# create a tietze object to reduce the presentation a bit
if not IsBound(aug.subgroupRelators) then
aug.subgroupRelators := RewriteSubgroupRelators( aug, aug.groupRelators);
fi;
# as the presentation might be rather long, we do not decode all secondary
# generators and their images, but will do it ``on the fly'' when
# rewriting.
aug:=CopiedAugmentedCosetTable(aug);
pres := PresentationAugmentedCosetTable( aug, "y",0# printlevel
,true) ;# intialize tracking before the `1or2' routine!
opt:=TzOptions(pres);
if ValueOption("expandLimit")<>fail then
opt.expandLimit:=ValueOption("expandLimit");
else
opt.expandLimit:=108; # do not grow too much.
fi;
if ValueOption("eliminationsLimit")<>fail then
opt.eliminationsLimit:=ValueOption("eliminationsLimit");
else
opt.eliminationsLimit:=20; # do not be too greedy
fi;
if ValueOption("lengthLimit")<>fail then
opt.lengthLimit:=ValueOption("lengthLimit");
else
opt.lengthLimit:=Int(3/2*pres!.tietze[TZ_TOTAL]); # not too big.
fi;
if ValueOption("generatorsLimit")<>fail then
opt.lengthLimit:=ValueOption("generatorsLimit");
fi;
TzOptions(pres).printLevel:=InfoLevel(InfoFpGroup);
TzEliminateRareOcurrences(pres,50);
TzGoGo(pres); # cleanup
# new free group
f:=FpGroupPresentation(pres,str);
# images for the old primary generators
aug.primaryImages:=Immutable(List(
TzImagesOldGens(pres){[1..Length(aug.primaryGeneratorWords)]},
i->MappedWord(i,GeneratorsOfPresentation(pres),GeneratorsOfGroup(f))));
TrySecondaryImages(aug);
# generator numbers of the new generators
w:=List(TzPreImagesNewGens(pres),
i->aug.treeNumbers[Position(OldGeneratorsOfPresentation(pres),i)]);
# and the corresponding words in the original group
w:=List(w,i->TreeRepresentedWord(aug.primaryGeneratorWords,aug.tree,i));
if not IsWord(One(u)) then
fam:=ElementsFamily(FamilyObj(u));
w:=List(w,i->ElementOfFpGroup(fam,i));
fi;
# write the homomorphism in terms of the image's free generators
# (so preimages are cheap)
# this object cannot test whether is is a proper mapping, so skip
# safety assertions that could be triggered by the construction process
f:=GroupHomomorphismByImagesNC(u,f,w,GeneratorsOfGroup(f):noassert);
# but give it `aug' as coset table, so we will use rewriting for images
SetCosetTableFpHom(f,aug);
SetIsBijective(f,true);
return f;
end);
InstallMethod(IsomorphismFpGroupByGeneratorsNC,"subgroups of fp group",
IsFamFamX,
[IsSubgroupFpGroup,IsList and IsMultiplicativeElementWithInverseCollection,
IsString],0,
function(u,gens,nam)
local aug,w,p,pres,f,fam,G,trace;
trace:=[];
if HasIsWholeFamily(u) and IsWholeFamily(u) and
IsIdenticalObj(gens,GeneratorsOfGroup(u)) then
return IdentityMapping(u);
fi;
# get an augmented coset table from the group. It must be compatible with
# `gens', so we must always use MTC.
# use new MTC
w:=FamilyObj(u)!.wholeGroup;
aug:=NEWTC_CosetEnumerator(FreeGeneratorsOfFpGroup(w),
RelatorsOfFpGroup(w),
List(gens,UnderlyingElement),
true,trace);
pres:=NEWTC_PresentationMTC(aug,1,nam);
if Length(GeneratorsOfPresentation(pres))>Length(gens) then
aug:=NEWTC_CosetEnumerator(FreeGeneratorsOfFpGroup(w),
RelatorsOfFpGroup(w),
List(gens,UnderlyingElement),
true,trace);
pres:=NEWTC_PresentationMTC(aug,0,nam);
fi;
Assert(0,Length(GeneratorsOfPresentation(pres))=Length(gens));
# new free group
f:=FpGroupPresentation(pres);
aug.homgens:=gens;
aug.homgenims:=GeneratorsOfGroup(f);
aug.primaryImages:=GeneratorsOfGroup(f);
aug.secondaryImages:=ShallowCopy(GeneratorsOfGroup(f));
f:=GroupHomomorphismByImagesNC(u,f,gens,GeneratorsOfGroup(f):noassert);
# tell f, that `aug' can be used as its coset table
SetCosetTableFpHom(f,aug);
SetIsBijective(f,true);
return f;
end);
#############################################################################
##
#F IsomorphismSimplifiedFpGroup(G)
##
##
InstallMethod(IsomorphismSimplifiedFpGroup,"using tietze transformations",
true,[IsSubgroupFpGroup],0,
function ( G )
local H, pres,map,mapi,opt;
# check the given argument to be a finitely presented group.
if not ( IsSubgroupFpGroup( G ) and IsGroupOfFamily( G ) ) then
Error( "argument must be a finitely presented group" );
fi;
# convert the given group presentation to a Tietze presentation.
pres := PresentationFpGroup( G, 0 );
# perform Tietze transformations.
opt:=TzOptions(pres);
if ValueOption("protected")<>fail then
opt.expandLimit:=ValueOption("protected");
fi;
opt.printLevel:=InfoLevel(InfoFpGroup);
TzInitGeneratorImages(pres);
if ValueOption("easy")=true then
# case of old `SimplifiedFpGroup`, use default strategy parameters
TzGo( pres );
else
# Somewhat tuned strategy parameters
if ValueOption("expandLimit")<>fail then
opt.expandLimit:=ValueOption("expandLimit");
else
opt.expandLimit:=120; # do not grow too much.
fi;
if ValueOption("eliminationsLimit")<>fail then
opt.eliminationsLimit:=ValueOption("eliminationsLimit");
else
opt.eliminationsLimit:=20; # do not be too greedy
fi;
if ValueOption("lengthLimit")<>fail then
opt.lengthLimit:=ValueOption("lengthLimit");
else
opt.lengthLimit:=Int(3*pres!.tietze[TZ_TOTAL]); # not too big.
fi;
TzGoGo( pres );
fi;
# reconvert the Tietze presentation to a group presentation.
H := FpGroupPresentation( pres );
if Length(GeneratorsOfGroup(H))>0 then
map:=List(TzImagesOldGens(pres),
i->MappedWord(i,GeneratorsOfPresentation(pres),
GeneratorsOfGroup(H)));
else
map:=List(TzImagesOldGens(pres),y->One(H));
fi;
map:=GroupHomomorphismByImagesNC(G,H,GeneratorsOfGroup(G),map);
mapi:=GroupHomomorphismByImagesNC(H,G,GeneratorsOfGroup(H),
List(TzPreImagesNewGens(pres),
i->MappedWord(i,OldGeneratorsOfPresentation(pres),
GeneratorsOfGroup(G))));
SetIsBijective(map,true);
SetInverseGeneralMapping(map,mapi);
SetInverseGeneralMapping(mapi,map);
ProcessEpimorphismToNewFpGroup(map);
return map;
end );
#############################################################################
##
#M SimplifiedFpGroup( <FpGroup> ) . . . . . . . . . simplify the FpGroup by
#M Tietze transformations
##
## `SimplifiedFpGroup' returns a group isomorphic to the given one with a
## presentation which has been tried to simplify via Tietze transformations.
##
InstallGlobalFunction( SimplifiedFpGroup, function ( G )
return Range(IsomorphismSimplifiedFpGroup(G:easy));
end);
#############################################################################
##
#M NaturalHomomorphismByNormalSubgroup(<G>,<N>)
##
InstallMethod(NaturalHomomorphismByNormalSubgroupOp,
"for subgroups of fp groups",IsIdenticalObj,
[IsSubgroupFpGroup, IsSubgroupFpGroup],0,
function(G,N)
local T,m;
# try to use rewriting if the index is not too big.
if IndexInWholeGroup(G)>1 and IndexInWholeGroup(G)<=1000
and HasGeneratorsOfGroup(N) and not
HasCosetTableInWholeGroup(N) then
T:=IsomorphismFpGroup(G);
return T*NaturalHomomorphismByNormalSubgroup(Image(T,G),Image(T,N));
fi;
if not HasCosetTableInWholeGroup(N) and not
IsSubgroupOfWholeGroupByQuotientRep(N) then
# try to compute a coset table
T:=TryCosetTableInWholeGroup(N:silent:=true);
if T=fail then
if not (HasIsWholeFamily(G) and IsWholeFamily(G)) then
TryNextMethod(); # can't do
fi;
# did not succeed - do the stupid thing
m:=CosetTableDefaultMaxLimit;
repeat
m:=m*1000;
T:=TryCosetTableInWholeGroup(N:silent:=true,max:=m);
until T<>fail;
fi;
fi;
return NaturalHomomorphismByNormalSubgroupNC(G,
AsSubgroupOfWholeGroupByQuotient(N));
end);
InstallMethod(NaturalHomomorphismByNormalSubgroupOp,
"for subgroups of fp groups by quotient rep.",IsIdenticalObj,
[IsSubgroupFpGroup,
IsSubgroupFpGroup and IsSubgroupOfWholeGroupByQuotientRep ],0,
function(G,N)
local Q,B,Ggens,gens,hom;
Q:=N!.quot;
Ggens:=GeneratorsOfGroup(G);
# generators of G in image
gens:=List(Ggens,elm->
MappedWord(UnderlyingElement(elm),
FreeGeneratorsOfWholeGroup(N),GeneratorsOfGroup(Q)));
B:=SubgroupNC(Q,gens);
hom:=NaturalHomomorphismByNormalSubgroupNC(B,N!.sub);
gens:=List(gens,i->ImageElm(hom,i));
hom:=GroupHomomorphismByImagesNC(G,Range(hom),Ggens,gens);
SetKernelOfMultiplicativeGeneralMapping(hom,N);
return hom;
end);
InstallMethod(NaturalHomomorphismByNormalSubgroupOp,
"trivial image fp case",IsIdenticalObj,
[IsSubgroupFpGroup,
IsSubgroupFpGroup and IsWholeFamily ],0,
function(G,N)
local Q,Ggens,gens,hom;
Ggens:=GeneratorsOfGroup(G);
# generators of G in image
gens:=List(Ggens,elm->()); # a new group is created
Q:=GroupWithGenerators(gens);
hom:=GroupHomomorphismByImagesNC(G,Q,Ggens,gens);
SetKernelOfMultiplicativeGeneralMapping(hom,N);
return hom;
end);
#########################################################
##
#M MaximalAbelianQuotient(<fp group>)
##
##
InstallMethod(MaximalAbelianQuotient,"whole fp group",
true, [IsSubgroupFpGroup and IsWholeFamily], 0,
function(f)
local m,s,g,i,j,rel,gen,img,fin,hom,gens;
# since f is the full group, exponent sums are with respect to its
# generators.
m:=List(RelatorsOfFpGroup(f),w->ExponentSums(w));
gen:=GeneratorsOfGroup(f);
g:= FreeGroup( Length( gen ) );
gen:=GeneratorsOfGroup(g);
rel:=[];
for i in [1..Length(gen)-1] do
for j in [i+1..Length(gen)] do
Add(rel, Comm(gen[i],gen[j]));
od;
od;
if Length(m)>0 then
s:=NormalFormIntMat(m,25); # 9+16: SNF with transforms, destructive
SetAbelianInvariants(f,AbelianInvariantsOfList(DiagonalOfMat(s.normal)));
if Length(m[1])>s.rank then
img:=[];
for i in [1..s.rank] do
Add(img,s.normal[i][i]);
Add(rel,g.(i)^s.normal[i][i]);
od;
Append(img,ListWithIdenticalEntries(Length(gen)-s.rank,0));
SetAbelianInvariants(f,img);
g:=g/rel;
fin:=false;
else
SetAbelianInvariants(f,DiagonalOfMat(s.normal));
g:=AbelianGroup(DiagonalOfMat(s.normal));
fin:=true;
fi;
gen:=GeneratorsOfGroup(g);
s:=s.coltrans;
img:=[];
for i in [1..Length(s)] do
m:=Identity(g);
for j in [1..Length(gen)] do
m:=m*gen[j]^s[i][j];
od;
Add(img,m);
od;
else
g:=g/rel;
fin:=Length(gen)=0;
img:=GeneratorsOfGroup(g);
SetAbelianInvariants(f,ListWithIdenticalEntries(Length(gen),0));
fi;
SetIsFinite(g,fin);
SetIsAbelian(g,true);
hom:=GroupHomomorphismByImagesNC(f,g,GeneratorsOfGroup(f),img);
SetIsSurjective(hom,true);
return hom;
end);
InstallMethod(MaximalAbelianQuotient,
"for subgroups of finitely presented groups",
true, [IsSubgroupFpGroup], 0,
function(U)
local phi,m;
phi:=IsomorphismFpGroup(U);
m:=MaximalAbelianQuotient(Image(phi));
SetAbelianInvariants(U,AbelianInvariants(Image(phi)));
return phi*MaximalAbelianQuotient(Image(phi));
end);
# u must be a subgroup of the image of home
InstallGlobalFunction(
LargerQuotientBySubgroupAbelianization,function(hom,u)
local v,ma,mau,a,gens,imgs,q,k,co,aiu,aiv,primes,irrel;
v:=PreImage(hom,u);
aiv:=AbelianInvariants(v);
aiu:=AbelianInvariants(u);
if aiu=aiv then
return fail;
fi;
# are there irrelevant primes?
primes:=Union(List(Union(aiu, aiv), PrimeDivisors));
irrel:=Filtered(primes,x->Filtered(aiv,z->IsInt(z/x))=
Filtered(aiu,z->IsInt(z/x)));
Info(InfoFpGroup,1,"Larger by factor ",Product(aiv)/Product(aiu),"\n");
ma:=MaximalAbelianQuotient(v);
mau:=MaximalAbelianQuotient(u);
a:=Image(ma);
k:=TrivialSubgroup(a);
for primes in irrel do
k:=ClosureGroup(k,GeneratorsOfGroup(SylowSubgroup(a,primes)));
od;
if Size(k)>1 then
ma:=ma*NaturalHomomorphismByNormalSubgroup(a,k);
a:=Image(ma);
k:=TrivialSubgroup(Image(mau));
for primes in irrel do
k:=ClosureGroup(k,GeneratorsOfGroup(SylowSubgroup(Image(mau),primes)));
od;
mau:=mau*NaturalHomomorphismByNormalSubgroup(Image(mau),k);
fi;
gens:=SmallGeneratingSet(a);
imgs:=List(gens,x->Image(mau,Image(hom,PreImagesRepresentative(ma,x))));
q:=GroupHomomorphismByImages(a,Image(mau),gens,imgs);
k:=KernelOfMultiplicativeGeneralMapping(q);
co:=ComplementClassesRepresentatives(a,k);
if Length(co)=0 then
co:=List(ConjugacyClassesSubgroups(a),Representative);
co:=Filtered(co,x->Size(Intersection(k,x))=1);
Sort(co,function(a,b) return Size(a)>Size(b);end);
fi;
Info(InfoFpGroup,2,"Degree larger ",Index(a,co[1]),"\n");
return PreImage(ma,co[1]);
end);
DeclareRepresentation("IsModuloPcgsFpGroupRep",
IsModuloPcgs and IsPcgsDefaultRep, [ "hom", "impcgs", "groups" ] );
InstallMethod(ModuloPcgs,"subgroups fp",true,
[IsSubgroupFpGroup,IsSubgroupFpGroup],0,
function(M,N)
local hom,pcgs,impcgs;
hom:=NaturalHomomorphismByNormalSubgroupNC(M,N);
hom:=hom*IsomorphismSpecialPcGroup(Image(hom,M));
impcgs:=FamilyPcgs(Image(hom,M));
pcgs:=PcgsByPcSequenceCons(IsPcgsDefaultRep,IsModuloPcgsFpGroupRep,
ElementsFamily(FamilyObj(M)),
List(impcgs,i->PreImagesRepresentative(hom,i)),
[]
);
pcgs!.hom:=hom;
pcgs!.impcgs:=impcgs;
pcgs!.groups:=[M,N];
if IsFiniteOrdersPcgs(impcgs) then
SetIsFiniteOrdersPcgs(pcgs,true);
fi;
if IsPrimeOrdersPcgs(impcgs) then
SetIsPrimeOrdersPcgs(pcgs,true);
fi;
return pcgs;
end);
InstallMethod(NumeratorOfModuloPcgs,"fp",true,[IsModuloPcgsFpGroupRep],0,
p->GeneratorsOfGroup(p!.groups[1]));
InstallMethod(DenominatorOfModuloPcgs,"fp",true,[IsModuloPcgsFpGroupRep],0,
p->GeneratorsOfGroup(p!.groups[2]));
InstallMethod(RelativeOrders,"fp",true,[IsModuloPcgsFpGroupRep],0,
p->RelativeOrders(p!.impcgs));
InstallMethod(RelativeOrderOfPcElement,"fp",IsCollsElms,
[IsModuloPcgsFpGroupRep,IsMultiplicativeElementWithInverse],0,
function(p,e)
return RelativeOrderOfPcElement(p!.impcgs,ImagesRepresentative(p!.hom,e));
end);
InstallMethod(ExponentsOfPcElement,"fp",IsCollsElms,
[IsModuloPcgsFpGroupRep,IsMultiplicativeElementWithInverse],0,
function(p,e)
return ExponentsOfPcElement(p!.impcgs,ImagesRepresentative(p!.hom,e));
end);
InstallMethod(EpimorphismFromFreeGroup,"general",true,
[IsGroup and HasGeneratorsOfGroup],0,
function(G)
local F,str;
str:=ValueOption("names");
if IsList(str) and ForAll(str,IsString) and
Length(str)=Length(GeneratorsOfGroup(G)) then
F:=FreeGroup(str);
else
if not IsString(str) then
str:="x";
fi;
F:=FreeGroup(Length(GeneratorsOfGroup(G)),str);
fi;
return
GroupHomomorphismByImagesNC(F,G,GeneratorsOfGroup(F),GeneratorsOfGroup(G));
end);
InstallGlobalFunction(ProcessEpimorphismToNewFpGroup,
function(hom)
local s,r,fam,fas,fpf,mapi;
if not (HasIsSurjective(hom) and IsSurjective(hom)) then
Info(InfoWarning,1,"fp eipimorphism is created in strange way, bail out");
return; # hom might be ill defined.
fi;
s:=Source(hom);
r:=Range(hom);
mapi:=MappingGeneratorsImages(hom);
if mapi[2]<>GeneratorsOfGroup(r) then
return; # the method does not apply here. One could try to deal with the
#extra generators separately, but that is too much work for what is
#intended as a minor hint.
fi;
s:=SubgroupNC(s,mapi[1]);
fam:=FamilyObj(One(r));
fas:=FamilyObj(One(s));
if IsPermCollection(s) or IsMatrixCollection(s)
or IsPcGroup(s) or CanEasilyCompareElements(s) then
# in the long run this should be the inverse of the source restricted
# mapping (or the corestricted inverse) but that does not work well with
# current homomorphism code, thus build new map.
#fpf:=InverseGeneralMapping(hom);
fpf:=GroupHomomorphismByImagesNC(r,s,mapi[2],mapi[1]);
elif IsFpGroup(s) and HasFPFaithHom(fas) then
#fpf:=InverseGeneralMapping(hom)*FPFaithHom(fas);
fpf:=GroupHomomorphismByImagesNC(r,s,mapi[2],List(mapi[1],x->Image(FPFaithHom(fas),x)));
else
fpf:=fail;
fi;
if fpf<>fail then
SetEpimorphismFromFreeGroup(ImagesSource(fpf),fpf);
SetFPFaithHom(fam,fpf);
SetFPFaithHom(r,fpf);
if IsPermGroup(s) then SetIsomorphismPermGroup(r,fpf);fi;
if IsPcGroup(s) then SetIsomorphismPcGroup(r,fpf);fi;
fi;
end);
#############################################################################
##
#E
##