############################################################################# ## ## This file is part of GAP, a system for computational discrete algebra. ## This file's authors include Alexander Hulpke, Soley Jonsdottir. ## ## 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 an implementation of the Cannon/Holt automorphism ## group algorithm: ## Automorphism group computation and isomorphism testing in finite groups. ## J. Symb. Comput. 35, No. 3, 241-267 (2003)
# call as big,perm,aut (the latter two can be groups or generator lists, # optional permiso. # returns two groups with corresponding generators
BindGlobal( "AGSRReducedGens", function(arg)
local big,bp,A,permgens,s,auts,sel,i,sub;
big:=arg[1];
bp:=arg[2];
A:=arg[3];
if IsGroup(bp) then
permgens:=GeneratorsOfGroup(bp); else
permgens:=bp;
bp:=SubgroupNC(Parent(big),permgens); fi;
SetSize(bp,Size(big)); if IsGroup(A) then
auts:=GeneratorsOfGroup(A); else
auts:=A;
A:=fail; fi; if Length(permgens)<>Length(auts) then Error("correspondence!");fi;
s:=SmallGeneratingSet(big); if Length(s)+2>=Length(permgens) then
return fail; fi;
# first try to reduce by dropping generators
sel:=[1..Length(permgens)]; for i in [1..Length(permgens)] do
sub:=SubgroupNC(Parent(big),permgens{Difference(sel,[i])}); if Size(sub)=Size(big) then
RemoveSet(sel,i);
bp:=sub; fi;
od; # good enough? if Length(s)+2<Length(sel) then if Length(arg)>3then
i:=InverseGeneralMapping(arg[4]); else
i:=GroupHomomorphismByImagesNC(big,Group(auts),permgens,auts); fi;
auts:=List(s,x->ImagesRepresentative(i,x));
bp:=SubgroupNC(Parent(big),s);
SetSize(bp,Size(big));
auts:=Group(auts); else
auts:=Group(auts{sel}); fi;
SetIsGroupOfAutomorphismsFiniteGroup(auts,true);
SetIsFinite(auts,true);
SetSize(auts,Size(bp)); if A<>fail then if HasInnerAutomorphismsAutomorphismGroup(A) then
SetInnerAutomorphismsAutomorphismGroup(auts,
InnerAutomorphismsAutomorphismGroup(A)); fi; if HasNiceMonomorphism(A) then
SetNiceMonomorphism(auts,NiceMonomorphism(A));
SetNiceObject(auts,NiceObject(A)); fi; fi;
return [bp,auts]; end );
# If M<=Frat(C_G(M)), try to find relators for C/M that in G evaluate to # generators of M and for which exponent sums are multiples of p. In this # case the values of the relators on pre-images in G do not depend on choice # of representatives and can be used to deduce the module automorphism # belonging to a factor group automorphism.
BindGlobal("AGSRFindRels",function(nat,newgens)
local C,M,p,all,gens,sub,q,hom,fp,rels,new,pre,i,free,cnt;
M:=KernelOfMultiplicativeGeneralMapping(nat);
C:=Centralizer(Source(nat),M); if not IsSubset(FrattiniSubgroup(C),M) then
return fail; fi;
p:=SmallestPrimeDivisor(Size(M)); all:=[]; if newgens=true then # so generators new
sub:=TrivialSubgroup(Image(nat)); while Size(sub)<Size(Image(nat)) do
sub:=ClosureGroup(sub,ImagesRepresentative(nat,Random(C)));
od; else
sub:=Image(nat,C); fi;
gens:=SmallGeneratingSet(sub);
free:=FreeGroup(Length(gens));
sub:=TrivialSubgroup(M);
cnt:=0; while Size(sub)<Size(M) do
q:=Group(gens);
SetSize(q,Size(Image(nat,C))); # use `ByGenerators` to force more random relators
hom:=IsomorphismFpGroupByGenerators(q,gens);
fp:=Range(hom);
rels:=Filtered(RelatorsOfFpGroup(fp),x->ForAll(ExponentSums(x),x->x mod p=0));
rels:=List(rels,x->ElementOfFpGroup(FamilyObj(One(fp)),x));
new:=RestrictedMapping(nat,C)*hom;
pre:=List(rels,x->PreImagesRepresentative(new,x)); for i in [1..Length(rels)] do if not pre[i] in sub then
Add(all,MappedWord(rels[i],
GeneratorsOfGroup(fp),GeneratorsOfGroup(free)));
sub:=ClosureGroup(sub,pre[i]); fi;
od;
cnt:=cnt+1; if cnt>5then return fail;fi;
od;
return rec(gens:=gens,free:=free,rels:=all); end);
BindGlobal("AGSRPrepareAutomLift",function(G,pcgs,nat)
local ocr,fphom,fpg,free,len,dim,tmp,L0,R,rels,mat,r,RS,i,g,v,cnt;
if Size(ocr.moduleauts)> # Finding the relations comes at a cost that needs to be plausible # with searching multiple times through the automorphism group. This # order bound is a heuristic that seems to be OK by magnitude. 100 then
cnt:=0;
repeat
ocr.trickrels:=AGSRFindRels(nat,cnt>3);
cnt:=cnt+1;
until ocr.trickrels<>fail or 2^cnt>100*Size(ocr.moduleauts); if ocr.trickrels=fail then Info(InfoMorph,1,"trickrels fails");fi; else
ocr.trickrels:=fail; fi;
# Initialize system.
len:=Length(ocr.generators);
dim:=Length(pcgs);
tmp := ocr.moduleMap( ocr.identity );
L0 := Concatenation( List( [ 1 .. len ], x -> tmp ) );
ConvertToVectorRep(L0,ocr.field);
R := ListWithIdenticalEntries( len * dim,Zero( ocr.field ) );
ConvertToVectorRep(R,ocr.field);
rels:=ocr.relators;
mat:=List([1..len*dim],x->[]); for i in mat do
ConvertToVectorRep(i,ocr.field);
od; for i in [1..Length(rels)] do
Info(InfoCoh,2," relation ", i, " (",Length(rels),")");
r:=1; for g in [1..len] do
RS:=OCEquationMatrix(ocr,rels[i],g); for v in RS do
Append(mat[r],v);
r:=r+1;
od;
od;
od;
ocr.matrix:=ImmutableMatrix(ocr.field,mat);
ocr.semiech:=ShallowCopy(SemiEchelonMatTransformation(ocr.matrix));
ocr.semiech.numrows:=NrRows(ocr.matrix);
return ocr; end);
# solve using the stored LR decomposition
BindGlobal("AGSRSolMat",function(sem,vec)
local i,vno,x,z,sol;
z := ZeroOfBaseDomain(sem.vectors);
sol := ListWithIdenticalEntries(sem.numrows,z);
ConvertToVectorRepNC(sol); for i in [1..Length(vec)] do
vno := sem.heads[i]; if vno <> 0then
x := vec[i]; if x <> z then
AddRowVector(vec, sem.vectors[vno], -x);
AddRowVector(sol, sem.coeffs[vno], x); fi; fi;
od; if IsZero(vec) then
return sol; else
return fail; fi; end);
############################################################################# ## #F OCEquationVectorAutom(<ocr>,<r>,<genimages>) ##
BindGlobal("OCEquationVectorAutom",function(ocr,r,genimages)
local n,i;
# If <r> has an entry 'conjugated' the records is no relator for a # presentation,but belongs to relation # (g_i n_i) ^ s_j =<r> # which is used to determinate normal complements. [i,j] is bound to # <conjugated>. if IsBound(r.conjugated) then
Error("not yet implemented"); fi;
n:=ocr.identity;
for i in [1 .. Length(r.generators)] do
n:=n*genimages[r.generators[i]]^r.powers[i];
od;
Assert(1,n in GroupByGenerators(NumeratorOfModuloPcgs(ocr.modulePcgs)));
return ShallowCopy(ocr.moduleMap(n));
end);
BindGlobal("AGSRAutomLift",function(ocr,nat,fhom,miso)
local v, rels, genimages, v1, psim, w, s, t, l, hom, i, e, j,ep,phom,enum;
v:=[];
rels:=ocr.relators;
genimages:=List(ocr.factorgens,i->MappedWord(
ImagesRepresentative(ocr.decomp,Image(fhom,i)),
GeneratorsOfGroup(ocr.free),
ocr.generators)); for i in [1..Length(rels)] do
v1:=OCEquationVectorAutom(ocr,rels[i],genimages);
Add(v,v1);
od;
#for ep in Enumerator(ocr.moduleauts) do if ocr.trickrels<>fail then # special case for M<=Frat(C_G(M)). Use special relators for factor that # allow to deduce corresponding module aut.
t:=ocr.trickrels;
phom:=IdentityMapping(ocr.moduleauts);
s:=List(t.gens,x->PreImagesRepresentative(nat,x));
l:=List(t.gens,x->PreImagesRepresentative(nat,ImagesRepresentative(fhom,x)));
s:=List(t.rels,x->MappedWord(x,GeneratorsOfGroup(t.free),s));
l:=List(t.rels,x->MappedWord(x,GeneratorsOfGroup(t.free),l));
s:=List(s,x->ExponentsOfPcElement(ocr.modulePcgs,x))*One(ocr.field);
l:=List(l,x->ExponentsOfPcElement(ocr.modulePcgs,x))*One(ocr.field);
Info(InfoMorph,5,"Deduced corresponding module automorphism"); if RankMat(l)<Length(l) then
return fail; fi;
enum:=[s^-1*l];
else
phom:=IsomorphismPermGroup(ocr.moduleauts);
enum:=Enumerator(Image(phom,ocr.moduleauts));
Info(InfoMorph,5,"Search through module automorphisms of size ",
Size(Image(phom))); fi; for ep in enum do
e:=PreImagesRepresentative(phom,ep);
psim:=e*miso;
psim:=psim^-1;
w:=-List(v,i->i*psim); #s:=SolutionMat(ocr.matrix,Concatenation(w));
s:=AGSRSolMat(ocr.semiech,Concatenation(w)); if s<>fail then
psim:=psim^-1;
t:=[];
ConvertToVectorRep(t,ocr.field);
l:=Length(ocr.modulePcgs); for i in [1..Length(genimages)] do
v1:=s{[(i-1)*l+1..(i*l)]}*psim; for j in [1..Length(v1)] do
t[(i-1)*l+j]:=v1[j];
od;
od;
s:=ocr.cocycleToList(t); for i in [1..Length(genimages)] do
genimages[i]:=genimages[i]*s[i];
od;
# later use NC version
hom:=GroupHomomorphismByImagesNC(ocr.group,ocr.group,
ocr.generators,genimages);
Assert(2,IsBijective(hom));
return hom; fi;
od;
return fail;
end);
# Find a larger subgroup that satisfies a condition, when testing # the condition can become expensive. # First try `SubgroupProperty`, but when it stalls attempt to find # minimal supergroups and prove that none of them satisfies.
BindGlobal("SubgroupConditionAboveAux",function(G,cond,S1,avoid)
local S,c,hom,q,a,b,i,t,have,ups,new,u,good,abort,clim,worked,pp,
cnt,locond,tstcnt,setupc,havetest;
setupc:=function()
hom:=NaturalHomomorphismByNormalSubgroupNC(G,u);
q:=Image(hom,G);
ups:=Image(hom,avoid); # aim for zuppos that cannot intersect avoid
c:=ConjugacyClasses(q);
c:=Filtered(c,x->IsPrimePowerInt(Order(Representative(x))) and
not Representative(x) in ups); # elements that do not have prime-order power in the subgroup avoid
c:=Filtered(c,x->not Representative(x)^
(Order(Representative(x))/SmallestPrimeDivisor(Order(Representative(x))))
in ups); # this also implies prime powers after the respective primes
SortBy(c,Size);
Info(InfoMorph,3,Length(c)," classes with ",Sum(c,Size)," subgroups"); end;
# Strategy: First try `SubgroupProperty` with a bailout limit, just in # case it finds (enough) elements. The bailout limit is set smaller if the # factor is solvable, as it will be cheaper to find complementing zuppos # in this case.
u:=Intersection(S,avoid); if IsNormal(G,u) and IsNormal(G,avoid)
and HasSolvableFactorGroup(G,u) then
setupc();
clim:=Minimum(Maximum(QuoInt(Sum(c,Size),4),10),1000); else
hom:=fail; # if less than 1/100 percent of elements succeed, assume close # to the subgroup has been found, and rather aim to prove there # will not be more.
c:=fail;
clim:=Maximum(QuoInt(IndexNC(G,S),10000),1000); fi;
# first try, how far `SubgroupProperty` goes
b:=0;
tstcnt:=0;
abort:=false;
havetest:=[];
if IsPermGroup(G) then
worked:=SubgroupProperty(G, function(elm) if abort then
return true; # are we bailing out since it behaves too badly? fi; # would it contribute to avoid outside S? if elm in avoid or
ForAny(Set(Factors(Order(elm))),e->elm^e in avoid and not elm^e in S) then # cannot be good
return false; fi;
tstcnt:=tstcnt+1;
AddSet(havetest,elm); if cond(elm) then
S:=ClosureGroup(S,elm); # remember
Info(InfoMorph,3,"New element ",IndexNC(G,S));
b:=0;
return true; else
b:=b+1; if b>clim then
abort:=true; fi;
return false; fi; end,S); else
worked:=S;
abort:=true; fi;
if abort=false then # we actually found the subgroup
Info(InfoMorph,2,"SubgroupProperty finds index ",IndexNC(G,worked));
return worked; fi;
Info(InfoMorph,2,"SubgroupProperty did ",tstcnt," tests and improved by ",
IndexNC(S,S1));
if not (IsNormal(G,u) and IsNormal(G,avoid)) then
Error("may only call if normalizing"); fi;
if c=fail then setupc();fi;
havetest:=Set(List(havetest,x->ImagesRepresentative(hom,x)));
clim:= # avoid writing down a permutation representation on more than 150000; #cosets, as it gets too memory expensive.
worked:=[]; # indicate whether class worked
# now run over all Zuppos (Conjugates of class representatives) in factor that do # not intersect (the image of) avoid. These, that saisfy, will span the correct subgroup.
i:=1; while worked<>fail and i<=Length(c) do
#should we abort? if Sum(c{[i..Length(c)]},Size)>IndexNC(G,S) and IndexNC(G,S)<=clim then
a:=Normalizer(G,S); # if the normalizer index is large, there will be many subgroups above if IndexNC(a,S)<200then
worked:=fail; # abort and go to other method fi; fi;
# if prime powers, the primes must have worked
a:=Representative(c[i]);
# no coprime power in earlier class?
have:=worked<>fail and
ForAll(Filtered([2..Order(a)-1],o->Gcd(o,Order(a))=1),
o->PositionProperty(c{[1..i-1]},x->a^o in x)=fail);
pp:=false; if have and not IsPrimeInt(Order(a)) then
pp:=SmallestPrimeDivisor(Order(a));
a:=a^(Order(a)/pp);
have:=worked[PositionProperty(c,x->a in x)]; fi;
if have then
have:=false;
a:=Representative(c[i]);
t:=Orbit(q,a);
t:=Filtered(t,x->not x in havetest);
cnt:=0; for b in t do
new:=PreImagesRepresentative(hom,b); if (pp=false or new^pp in S) and locond(new) then
S:=ClosureGroup(S,new);
have:=true;
cnt:=cnt+1; fi;
od;
Info(InfoMorph,4,"Did class ",i," order =",Order(Representative(c[i])), ", len=",Size(c[i])," newtest=", Length(t), " found ",cnt,": ",Size(S)); fi; if worked<>fail then Add(worked,have);fi;
i:=i+1;
od;
if worked=fail then # still need to test
Info(InfoMorph,2,"Go to blocks test");
if Index(G,S)>clim then Error("clim"); fi;
a:=SmallGeneratingSet(G); if Length(GeneratorsOfGroup(G))>Length(a) then
b:=Size(G);
G:=Group(a);
SetSize(G,b); fi;
repeat
good:=false;
u:=Filtered(GeneratorsOfGroup(G),x->cond(x) and not x in S); for i in u do
S:=ClosureGroup(S,i);
od;
# try to prove no supergroup works
t:=RightTransversal(G,S:noascendingchain); # don't try to be clever in # decomposing transversal, as this could be hard
a:=Action(G,t,OnRight); # coset action, don't need homomorphism
b:=RepresentativesMinimalBlocks(a,MovedPoints(a));
Info(InfoMorph,3,"Above are ",Length(b)," blocks");
if IsPermGroup(G) and Length(b)*10>IndexNC(G,S) then # there are too many blocks. Direct test is cheaper!
S:=SubgroupProperty(G,locond,S); else
for i in [1..Length(b)] do
CompletionBar(InfoMorph,3,"SubgroupsAboveBlocks ",i/Length(b));
c:=First(b[i],x->x>1); if cond(t[c]) then
S:=ClosureGroup(S,t[c]);
good:=true; fi;
od;
CompletionBar(InfoMorph,3,"SubgroupsAboveBlocks ",false); fi;
until good=false;
fi;
Info(InfoMorph,3,"Did ",tstcnt," tests, grow by ",Index(S,S1));
return S;
end);
# Same syntax as `SubgroupProperty`, but assumption that the tests are # expensive. Thus minimize number of tests by doing more group calculations
InstallGlobalFunction(SubgroupConditionAbove,function(G,cond,Sorig)
local cs,nr,u,no,un,S,rad,res,ise,uno;
# first, try to find a few elements that work (and catch the case that the # subgroup has small index. Ensure we test at least once for each possible # step.
nr:=2^LogInt(Size(G),1000)+Length(Factors(Size(G)));
u:=[]; if IsPermGroup(G) then
S:=SubgroupProperty(G, function(elm) if nr<0then return true;fi;
nr:=nr-1; if cond(elm) then
Add(u,elm);
return true; else
return false; fi; end,Sorig); else
u:=Filtered(GeneratorsOfGroup(G),cond);
nr:=-1; fi;
if nr>=0then# succeeded (small index case)
return S; elif Length(u)>0then
S:=ClosureGroup(Sorig,u); else
S:=Sorig; fi;
# now build along composition series
rad:=SolvableRadical(G); # composition series through perfect residuum seems to work better # Possible reason: Simple composition factors arise as inner # automorphisms. Moving the perfect residuum (if smaller) in the series, # thus increases the chance that much of the subgroup is found early (and # does not need to be found through search through complements)
res:=PerfectResiduum(G);
nr:=[Core(G,S),NormalClosure(G,S)]; if Size(nr[1])=Size(nr[2]) then nr:=nr{[1]};fi;
# refine with perfect residuum
no:=Intersection(nr[1],res); if not no in nr then nr:=Concatenation([no],nr);fi; if Length(nr)>1then
no:=ClosureGroup(Last(nr),res); if not no in nr then Add(nr,no);fi; fi;
cs:=CompositionSeriesThrough(G,nr);
nr:=First([1..Length(cs)],x->IsSubset(S,cs[x]));
uno:=false; while nr>1do
nr:=nr-1; if IsSubset(cs[nr+1],S) then
u:=S;
ise:=u; if uno=false then # if the group is huge use the radical-based normalizer if Size(rad)>10^13 and IndexNC(G,ise)>10^5then
Info(InfoMorph,4,"Radical-based Normalizer:",IndexNC(G,u));
uno:=NormalizerViaRadical(G,ise); else
Info(InfoMorph,4,"Ordinary Normalizer:",IndexNC(G,u));
uno:=Normalizer(G,ise); fi; fi;
no:=Intersection(uno,cs[nr]); else
# if the group is huge use the radical-based normalizer if Size(Intersection(rad,cs[nr]))>10^13 and IndexNC(cs[nr],ise)>10^5then
Info(InfoMorph,4,"Radical-based Normalizer:",IndexNC(cs[nr],ise), " of ",Index(cs[nr],cs[nr+1]));
no:=NormalizerViaRadical(cs[nr],ise); else
Info(InfoMorph,4,"Ordinary Normalizer:",IndexNC(cs[nr],ise), " of ",Index(cs[nr],cs[nr+1]));
no:=Normalizer(cs[nr],ise); fi; fi;
un:=SubgroupConditionAboveAux(no,cond,u,Intersection(no,cs[nr+1]));
Info(InfoMorph,2, "Step ",nr,": ",IndexNC(cs[nr],cs[nr+1])," to ",IndexNC(un,u)); if not IsSubset(S,un) then
S:=ClosureGroup(S,un);
uno:=false; fi;
od;
return S; end);
# find classes of normal subgroups
BindGlobal("AGSRNormalSubgroupClasses",function(G)
local fp,n,pat,pools,i,sel; # fingerprint
fp:=function(x)
local l; if ID_AVAILABLE(Size(x)) <> fail
and ValueOption(NO_PRECOMPUTED_DATA_OPTION)<>true then
return IdGroup(x); fi;
l:=[Size(x)];
Add(l,Collected(List(ConjugacyClasses(x),
y->[Order(Representative(y)),Size(y)])));
Add(l,AbelianInvariants(x));
return l; end;
n:=ValueOption("directs"); if n<>fail then # avoid large number of normals in direct product
n:=Concatenation(List(n,
x->Filtered(NormalSubgroups(x),y->Size(y)>1 and Size(y)<Size(x)))); else
n:=NormalSubgroups(G); fi;
pat:=List(n,fp);
pools:=[]; for i in Set(pat) do
sel:=Filtered([1..Length(n)],x->pat[x]=i);
Add(pools,n{sel});
od;
return pools; end);
# form a characterististic series through radical with elab factors
BindGlobal("AGSRCharacteristicSeries",function(G,r)
local somechar,d,i,j,u,v;
d:=Filtered(StructuralSeriesOfGroup(G),x->IsSubset(r,x)); # refine
d:=RefinedSubnormalSeries(d,Centre(G));
d:=RefinedSubnormalSeries(d,Centre(r));
somechar:=ValueOption("someCharacteristics"); if somechar<>fail then if IsRecord(somechar) then
somechar:=somechar.subgroups; fi; for i in somechar do
d:=RefinedSubnormalSeries(d,i);
od; fi;
if not ForAll([1..Length(d)-1],
x->HasElementaryAbelianFactorGroup(d[x],d[x+1])) then for i in PrimeDivisors(Size(r)) do
u:=PCore(r,i); if Size(u)>1then
d:=RefinedSubnormalSeries(d,u);
j:=1;
repeat
v:=Agemo(u,i,j); if Size(v)>1then
d:=RefinedSubnormalSeries(d,v); fi;
j:=j+1;
until Size(v)=1;
j:=1;
repeat if Size(u)>=2^24then
v:=u; # bail out as method for `Omega` will do so. else
v:=Omega(u,i,j); if Size(v)<Size(u) then
d:=RefinedSubnormalSeries(d,v); fi;
j:=j+1; fi;
until Size(v)=Size(u); fi;
od; fi;
Assert(1,ForAll([1..Length(d)-1],x->Size(d[x])<>Size(d[x+1])));
return d; end);
# main automorphism method -- currently still using factor groups, but # nevertheless faster..
# option somechar may be a list of characteristic subgroups, or a record with # component subgroups, orbits
BindGlobal("AutomGrpSR",function(G)
local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
b,fratsim,AQ,OQ,Zm,D,innC,oneC,imgs,C,maut,innB,tmpAut,imM,a,A,B,
cond,sub,AQI,AQP,AQiso,rf,res,resperm,proj,Aperm,Apa,precond,ac,
comiso,extra,mo,rada,makeaqiso,ind,lastperm,actbase,somechar,stablim,
scharorb,asAutom,jorb,jorpo,substb,isBadPermrep,ma,nosucl,nosuf,rlgf;
# criterion for when to force degree reduction
isBadPermrep:=function(g)
return NrMovedPoints(g)^3>Size(g)*Index(g,DerivedSubgroup(g)); end;
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 2
:)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
Info(s=java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 29 if HasIsomorphismPermGroup(AQ) then
AQiso:=IsomorphismPermGroup(AQ); elif HasNiceMonomorphism(AQ) and IsPermGroup(Range(NiceMonomorphism(AQ))) then
AQiso:=NiceMonomorphism(AQ:autactbase:=fail); else
a:=Filtered(GeneratorsOfGroup(AQ),HasConjugatorOfConjugatorIsomorphism); if Length(a)>2then
a:=List(a,ConjugatorOfConjugatorIsomorphism);
b:=SmallGeneratingSet(Group(a)); if Length(b)<Length(a) then
a:=List(b,x->ConjugatorAutomorphism(Source(AQ.1),x));
b:=(AQ),x> java.lang.StringIndexOutOfBoundsException: Range [88, 87) out of bounds for length 92
:Concatenation,b;
=java.lang.StringIndexOutOfBoundsException: Range [53, 48) out of bounds for length 53
:=(aOne);
SetInnerAutomorphismsAutomorphismGroupAQ
java.lang.StringIndexOutOfBoundsException: Range [49, 46) out of bounds for length 56 fi; fi fi
AQiso:=# generators of M and for which exponent sums are multiples# case the values of the relators on pre-images in G do not depend on choice
:java.lang.StringIndexOutOfBoundsException: Range [45, 41) out of bounds for length 47
AQiso:=IsomorphismPermGroup(AQ); fi; fi;
AQP:=Image(AQiso,AQ);
I3" of "Size(Q),deg:,NrMovedPoints();
GeneratorsOfGroup)java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 72
a:AQPAQP,AQ,)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43 if a>
java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 18
AQ=[] fi; fi;
# force degree down if java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 29
fphom:=java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 31 if NrMovedPointsImagea) then
Info(InfoMorph,3,"Permdegree reduced ",
NrMovedPointsAQP)">,NrMovedPoints(Image();
AQiso:=AQiso*a;
b:=Image(a,AQP);
()Length(GeneratorsOfGroup(AQP) then
b:= LinearActionLayer(ocr.ocr.modulePcgs)ocr;
SetSize(b if Size(cr.moduleauts)> fi;
AQP:=b; fi100
ocrtjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 45
;
.factorgens=ocr(,);
local no,same,sz,ac,i,sub;
same:=true;
repeat
szocr.java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 48 if gp)/>
(gp,)java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
()SizeAperm no<(p)
m)java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31 fi; else
java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 18
;
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
:AscendingChain(pAperm)
ListUnionList(,GeneratorsOfGroup),ond; # try generators... if Size(Aperm)>sz then
ac:=Unique(List"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 41 fi
i:if vno>0 java.lang.StringIndexOutOfBoundsException: Range [20, 21) out of bounds for length 20
java.lang.StringIndexOutOfBoundsException: Range [24, 20) out of bounds for length 46
; fi; #F OCEquationVectorAutom(<ocr>,<r>,<genimages>)
SubgroupProperty , fi;
same=(Aperm; if not same then
Info(InfoMorph,3,"stablim "remaining"Size()/Aperm) for [. r.]do
until same;
returnjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 end;
ff:[;
rff.java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 16
rlgf:=LGFirst(SpecialPcgs(r));
# find series through r
somechar:=ValueOption("someCharacteristics");
# derived and then primes and then elementary abelian
d:=(series") if d=fail ;
d:=AGSRCharacteristicSeries(G
ocr>then
java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
d:=hallowCopy(d;
SortBy(,) # in case reversed order.... fi
scharorb:=fail; if s:=List(sx-xponentsOfPcElement(cr.modulePcgs,x)*ocr.; if IsRecord(somechar) then if so)then
scharorbifRankMat(l<(l) fi:[^1*]
somechar=somechar.java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35 fi; fi;
u:=1; # last group in the series to be used
i:=2;
i<()java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
p:IndexNC([+]d[u); # should we skip group i? if (,cr.field);
d[+,d[u]) then
d:=Concatenation(d{[1..i-1]},d{[i+1..Length(d)]}); # i stays the same, as it now is the next subgroup else
uu=i;;
i:=i+1; fi;
od;
# avoid small central subgroups, as the factor will be hard to represent
u:=Centre if (u)1then
p:=SmallestPrimeDivisor( 2IsBijective()
u:=SylowSubgroup(u,p)o
ujava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
i:=Position(d,u);# the condition can become expensive. if i<>fail and# minimal supergroups and prove that none of them satisfies.BindGlobal(SubgroupConditionAboveAux"function(,S1,
d=dUnion(]i.}
=java.lang.StringIndexOutOfBoundsException: Range [49, 46) out of bounds for length 52
,xIsPrimePowerInt((java.lang.StringIndexOutOfBoundsException: Range [58, 57) out of bounds for length 66
ser:=[TrivialSubgroup(G)]; for i d{2.Length(d]} do
u:i in ) for p in PrimeDivisors(Size(i)/Size(u)) do
bd:=PValuation(Size(i)/Size(u),p); # max p-exponent
u:=ClosureSubgroup(u,SylowSubgroup(i,p));
v:=Last(ser);
lementaryAbelianFactorGroup,
gensS=java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
it finds (enough) elements. limit is smallerif
e:=p^(Maximum(e)-1); for j in gensu=Intersection(,avoid)java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
v:=ClosureSubgroup(v,j^e);
;
Add(ser,v);
od;
Add(ser,u);
od;
od;
rada:=fail;
fi
hom:=ff.factorhom;
Q try,how far`ubgroupProperty`goes
ermGroupQ and Q)/SizeQ)*Size(Socle(Q))
S(SizeQ) # just in case the radical factor hom is inherited.ifIsPermGroup() then
Q: # are we bailing out since it behaves too badly?
Info( elm inavoid or " -> ",NrMovedPoints(Range(Q)));
java.lang.StringIndexOutOfBoundsException: Range [10, 5) out of bounds for length 23
Q:=Image(hom,G); fi;
sesSolG);
AQ:=AutomorphismGroupFittingFree(Q:someCharacteristics:=fail);
AQI:=java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 17
lastperm # preseed natural homs in ascending form (as the largest one might help fi;
i Length()Length(ser)-1..2] do
od;
i:=1; while;
(AQ,->S(x)SizeQ);
t step is
lhom=hom;
OQ:oError(" if;
(,List(ser""i;
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 23
M:=ser[i];
N:=seri+1;
hom:=NaturalHomomorphismByNormalSubgroup(G,N);
Q:=Image(hom,G); # degree reduction called for? if IsPermGroup(Q) and Size(N)>1
and (isBadPermrep(Q) java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
qjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 if NrMovedPoints(Q)>1000
and NrMovedPoints(Q :=;
q:=SmallerDegreePermutationRepresentation
; if NrMovedPoints(Range(q))<NrMovedPoints(aNormalizer;
java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 72
worked=; abortand toother method
hom:=hom*java.lang.StringIndexOutOfBoundsException: Range [19, 20) out of bounds for length 0
Q fi; fi
# inherit radical factor map
pp=false;
List(GeneratorsOfGroup(G),x->ImagesRepresentative:=
List(GeneratorsOfGroup)x>(ff.factorhom,x);
b:=Image(hom,ffjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
:java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 30
AddNaturalHomomorphismsPoolcnt:0;
# Use known maximals for Frattini for j in ma do
D havetrue; if ;
b:=Core (InfoMorph,," class ",i," order=,Order(Representative([i)) fi;
od;
SetIsNilpotentGroup(b,true);
worked> Add(orked,;
# M-factor
Mim#still need totest
MPcgs:=Pcgs(Mim);
q:=(InfoMorph,2,"Go to blocks" java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 42
G(G))>a
List G:=Group(a);
AddNaturalHomomorphismsPool,im,q;
mo:=GModuleByMats(Q),),GF(RelativeOrders(MPcgs)[1])); # is the extension split?
:OneCocyclesQ,im
:ocr.java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34 ifnotsplit java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23 # test: Semisimple and Frattini
b=.m;
fratsim:=Length(baActionG,java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69 if not fratsim then
:List(,x>(om,x))); for j in b G Length)10IndexNC,)then
N:= S=ubgroupProperty(G,locond,S);
od; # insert in series for j in [Length(ser),Length(ser)-1..i+1] do
ser[j+1]:=ser[j];
od
ser[i+1]:=N;
Info( ([] java.lang.StringIndexOutOfBoundsException: Range [28, 29) out of bounds for length 28 elseod;
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
fratsim if not Info3" ,,"tests,grow by,ndex(,S1);
N:=Intersection(FrattiniSubgroup(Q),Mim); # insert for j in [Length(ser),Length(ser)-1..i+1] do
ser[j+1]:=ser[j];
od;
ser[i+1]:=PreImage(hom,N);
Info(InfoMorph,2,"insert2"); fi;
N:=ser[# expensive. Thus minimize number of tests by doing more group calculations fi; if
and # subgroup has small
Info :2^(()1000+Factors(G)
function(elm
NiceMonomorphism(rada:autactbase:=fail,someCharacteristics
=StabilizerradaN)java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
fi;
until split or fratsim;
Use cocycles
b:=BasisVectorsif >= then# succeeded (small index case)
# find D
Zm:=PreImage(q,Centre(OQ));
D:=Centralizer(Zm,Mim);
D:=List # thus increases the chance that much of the subgroup is found early (and # does not need to be found through search through complements)
D: :=CoreGS),S]
();
D:=Filtered(D,x->x<>0*x);
C:=[]; if Size(Group(ocr.generators))<Size(Q) then
extra:=MPcgs; else
extra:=[]; fi; for j in b do
oneC := ocr.cocycleToList( j );
:List[..Length.enerators],i>ocr.generators[i]*[])
oneC:= :First(1..engthc)x>(Scsx])
Assert(2,IsBijective(oneC));
Add,oneC;
od;
B:=[];
if lastperm<>fail then
AQiso:=lastperm;
=(java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 79 else
makeaqiso(); fi;
if split then
:=java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 40 find noninner of B
innB:=List(SmallGeneratingSet(Zm),z->InnerAutomorphism(Q,z));
innB:=Group(One(DefaultFieldOfMatrixGroup(maut))*
List(innB,inn->List(MPcgs,m->java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 64
tmpAut:=SubgroupNC(maut,Filtered(GeneratorsOfGroup(maut),aut->not java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 11
gens:=GeneratorsOfGroup(ocr.complement); for a in GeneratorsOfGroup(u:Intersection([nr]S);
ise:=(s[nr1,u;
#java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
Assert(IsBijective());
Add(B, Info(InfoMo(,4,IndexNC([nr,)java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
od;
# test condition for lifting, also add corresponding automorphismInfo(InfoMorph,4,Ordinary Normalizer:",IndexNC(cs[nr],ise), " of "Indexcsnr],nr+)java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 7
mo:=GModuleByMats:=(SmallGeneratingSetno)
cond:=java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
local aut,newgens,mo2, Info(, ifpermin
return if not IsSubset(S)then
java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length 17
aut:;
newgens:# find classes of normal subgroups
java.lang.StringIndexOutOfBoundsException: Range [34, 30) out of bounds for length 69
mo2and)> java.lang.StringIndexOutOfBoundsException: Range [60, 61) out of bounds for length 60
iso=MTX.IsomorphismModules(mo,o2;
=
return false; else # build associated auto
a:=GroupHomomorphismByImagesNCend;
Concatenation(newgens,
List(MPcgs,x->java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
(ExponentsOfPcElement(MPcgs,x)*One(mo.field))*iso ))));
Assert(2,IsBijective(a));
Add(A,a);
Add(Apa,perm)pat=List(nfp);
Aperm:=ClosureGroup(Aperm,perm);
return true; fi; end;
else # there is no B in the nonsplit case
B:=[];
ocr:=AGSRPrepareAutomLift( Q, java.lang.StringIndexOutOfBoundsException: Range [0, 41) out of bounds for length 25
java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 41
local aut,newgens somechar=ValueOption(") if permif IsRecordsomechar)then
return true; fi;
aut:=PreImagesRepresentative(AQiso,perm);
newgens::=ist(GeneratorsOfGroup(),
od;
mo2:=:=GModuleByMats(LinearActionLayer(newgens,MPcgs),mo.field);
return MTX.IsomorphismModules(mo,mo2)<java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0 endx>HasElementaryAbelianFactorGroup(dx],x+]) then
java.lang.StringIndexOutOfBoundsException: Range [20, 10) out of bounds for length 26
local aut,newgens,mo2 :RefinedSubnormalSeries(u); if perm in Aperm j=1
true; fi;
aut:=PreImagesRepresentative(AQiso,perm);
newgens:=List(GeneratorsOfGroup(Q),
x->PreImagesRepresentative(q,Image(aut,ImagesRepresentative(q,x))));
mo2:=GModuleByMats(LinearActionLayer(newgens,MPcgs),mo.field);
iso:=MTX.IsomorphismModules(java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 13 if iso=failuntil Sizev)=1java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
return false; else # build associated auto
a:=AGSRAutomLift(ocr,q,aut,iso); a:=AGSRAutomLift(ocr,q,autjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 28 if a=fail then #Print("test failed\n");
return false; else
Add Size(v=(u;
Add(Apaod;
Aperm:=ClosureGroup(Aperm,perm); #Print("test succeeded\n");
return true; fi; fi; end
fi;
# find A using the set condition
# option somechar may be a list of characteristic subgroups, or a record with
:=];
# note donot java.lang.StringIndexOutOfBoundsException: Range [38, 29) out of bounds for length 66
java.lang.StringIndexOutOfBoundsException: Range [48, 9) out of bounds for length 54
x->ImagesRepresentative(AQiso,x)));
# try to find some further generators if Size(AQP)/Size(Aperm)>100then for j ((AQP) do
cond(j;
od; for j in GeneratorsOfGroup(AQP) do
cond(j);
od;
makeaqiso=)
sub:=AQP;
precond<failand not (GeneratorsOfGroup(ub)precond) then # compatible pairs condition
sub:=SubgroupProperty(,,) fi;
if IndexNC(java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8 # try to find characteristic subgroupsifLength()2then
Info(InfoMorph,2,"Use normal :=SmallGeneratingSet(Group(a))
=failthen:java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 66
nosuf:=List(nosucl,x-> a:=Concatenation(a,b)
nosuf:=Filtered(nosuf,x->Sizex1]> Size(x[1]<OQ);
SortBy(nosuf,Length); for j in nosuf do
# stabilize class
k:=SetInnerAutomorphismsAuto(Qb;
ac:=OrbitStabilizerAlgorithm(sub,false,java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 11
(AQiso,x)),
rec(pnt:=j,
act:= function(set, else #local phi; #phi:=PreImagesRepresentative(AQiso,perm);
eturnSet(List(set,x->Image(phi,x))); end,
onlystab:=true));
Info(InfoMorph3" "subac; if Size(ac.stabilizer)<Size(sub) then
subifLengthGAQP)=engthA) then fi;
od; fi;
jfi;
Info(InfoMorph,2,"start search ",IndexNC(sub,Aperm));
sub:=SubgroupConditionAbove(sub,cond,Aperm);
Info(InfoMorph # force degree down
# Note: Aperm is larger than what is generated by Apa
j:=; while SizeAperm)Size(sub) do
ac:=InnerAutomorphism(OQ,Image(q,GeneratorsOfGroup(Q)[j]));
k:= if (mage()(AQP) then if not k in Aperm then
Add(Apa,k);
Aperm:=ClosureGroupNrMovedPoints(AQP),->,(mage())java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
Add,QG(Q[])java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60 fi;
j:=j+1;
od;
# remove redundant generators. Note that Aperm could be to big, thus # make group from Apa
ac:=AGSRReducedGens(SubgroupNC(Parent(sub),Apa),Apa,A); if ac<>fail then
Apa:=GeneratorsOfGroup(ac[1]);
A:=GeneratorsOfGroup(ac[2]); fi;
Info(InfoMorph,2,"Lift Index ",Size(AQP)/Size(sub));
# now make the new automorphism group
innB:=List(SmallGeneratingSet(Q),x- b=roupList(()x>java.lang.StringIndexOutOfBoundsException: Range [72, 70) out of bounds for length 78
gens:=ShallowCopy(innB);
SetSize(,AQP); "+",Length
Append(gens,java.lang.StringIndexOutOfBoundsException: Range [6, 17) out of bounds for length 9
Append
(java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 19
Assert(2,ForAll(gens,IsBijective)); for j in gens do
SetIsBijective(j,true);
od;
A:=Group(gens);
SetIsAutomorphismGroup(A,true Q:Image(om,G);
if IsPermGroup(Q) and NrMovedPoints(Q)(Q*(Socle(Q)
SetIsFinite(A,true);
AQI:SubgroupNC(,innB);
SetInnerAutomorphismsAutomorphismGroup(A,AQI);
AQ:=A;
Sizejava.lang.StringIndexOutOfBoundsException: Range [56, 48) out of bounds for length 56
(")>true then
makeaqiso();
ma:=Ma(G)
A:=AQ; fi; else
A.:makeaqiso; fi;
# do we use induced radical automorphisms to help next step? if Size(java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 5
Size(A)>10^8 #( ## potentially large GL #Size(GL(Length(MPcgs),RelativeOrders(MPcgs)[1]))>10^10 and from B/C-bit
# ## do so for all factors
+]
and ValueOptionnoradicalaut")>rue then
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 if IsElementaryAbelian(r) and Size(r) IsPermGroup(Q) and Size(N)>1
B:=Pcgs(r);
rf:=GF(RelativeOrders(B)[1]);
indQNrMovedPointsImageq)then
ind:=Listfi
java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 38
ind:=java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 11
:List(GeneratorsOfGroup(nd)-
GroupHomomorphismByImagesNC(r,r,B,List( GroupHomomorphismByImagesNC(r,r,B,List(x,y->m))java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
rada:==radical
AddNaturalHomomorphismsPool(Q,b
ase:failsomeCharacteristicsf) else
ind=IsomorphismPcGroup)
:AutomorphismGroup(Image,r)someCharacteristics:=ail,utactbase:fail); # we only consider those homomorphism that stabilize the series we use for;
GeneratorsOfGroup)x>magex,)>k)then
Info(InfoMorph,3,"radical java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 0
NiceMonomorphism(rada:autactbase :=roupHomomorphismByImagesNC(,,
SetIsGroupOfAutomorphismsFiniteGroup(eneratorsOfGroupG,x->(hom,),
rada: GeneratorsOfGroup)x>ImagesRepresentativex)) fi;
od; # move back to bad degree
rada:=Group(List(GeneratorsOfGroup(rada),
x-> InducedAutomorphism(InverseGeneralMapping(ind),x))); fi; fi;
rf:=Image(hom,r);
Info(InfoMorph,2,"Use java.lang.StringIndexOutOfBoundsException: Range [8, 35) out of bounds for length 21
makeaqiso(St(,,;
B:=MappingGeneratorsImages(AQiso);
res:=List(B[1until or;
GroupHomomorphismByImagesNC(rf,rf,
G(-(y))java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
ind:=[]; for j in GeneratorsOfGroup:List(,d->([.Length(ocr.],i-ocrgenerators[]-1i);
k:=GroupHomomorphismByImagesNC(rf,rf,
GeneratorsOfGroup(rf),
orsOfGroup(rf),
y-ImagesRepresentative(hom,ImagesRepresentative(j,
PreImagesRepresentative(hom,y)))));
Assert(2,IsBijective(k));
Add(ind,k);
od;
C:Group(Unique(Concatenation));
SetIsFinite(C,true);
SetIsGroupOfAutomorphismsFiniteGroup(C,true);
Size=
res:=SubgroupNC(C,res);
ind:=SubgroupNC(C,ind); # this should now go via the niceo of C
indQConcatenationocr.java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 107
:fail,fail;
ind:=Intersection(res,ind); # only those we care about
if Size(ind)<ize(res)then # reduce to subgroup that induces valid automorphisms
Info(InfoMorph,1java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7 # find noninner of B
proj=AQPImage(esperm)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61
,,>ImagesRepresentative,
C:=PreImage(proj,Image(resperm,ind));
C:=List(SmallGeneratingSet(C),x->PreImagesRepresentative(AQiso,x));
AQ:=Group(C);
SetIsFinite(AQ,true);
SetIsGroupOfAutomorphismsFiniteGroup(AQ,true);
makeaqiso(); fi;
# # hook for using existing characteristics to reduce for next step if somechar<>fail then
u:=Filtered(Unique(List(somechar,x->Image(hom,x))),x->Size(x)>1);
u:=Filtered(u,s->ForAny(GeneratorsOfGroup(AQ),h->Image(h,s)<>s));
SortBy(u,Size);
Info(InfoMorph,1,"Forced characteristics ",List(u,Size));
if scharorb<>fail then # these are subgroups for which certain orbits must be stabilized.
C:=List(Reversed(scharorb),x->List(x,y->Image(hom,y)));
C:=Filtered(C,x->Size(x[1])>1 and Size(x[1])<Size(Q));
Info(InfoMorph,1,"Forced orbits ",List(C,x->Size(x[1])));
Append(u,C); fi;
if Length(u)>0then
C:=MappingGeneratorsImages(AQiso); if C[2]<>GeneratorsOfGroup(AQP) then
C:=[List(GeneratorsOfGroup(AQP),
x->PreImagesRepresentative(AQiso,x)),
GeneratorsOfGroup(AQP)]; fi; for j in u do if IsList(j) then # stabilizer set of subgroups
jorb:=ShallowCopy(Orbit(AQP,j[1],C[2],C[1],asAutom));
jorpo:=[Position(jorb,j[1]),Position(jorb,j[2])]; if jorpo[2]=fail then
Append(jorb,Orbit(AQP,j[1],C[2],C[1],asAutom));
jorpo[2]:=Position(jorb,j[2]); fi; if Length(jorb)>Length(j) then
B:=ActionHomomorphism(AQP,jorb,C[2],C[1],asAutom);
substb:=Group(List(C[2],x->ImagesRepresentative(B,x)),());
substb:=Stabilizer(substb,Set(jorpo),OnSets);
substb:=PreImage(B,substb);
Info(InfoMorph,2,"Stabilize characteristic orbit ",Size(j[1]), " :",Size(AQP)/Size(substb) ); else
substb:=AQP; fi;
else
substb:=Stabilizer(AQP,j,C[2],C[1],asAutom);
Info(InfoMorph,2,"Stabilize characteristic subgroup ",Size(j), " :",Size(AQP)/Size(substb) ); fi; if Size(substb)<Size(AQP) then
B:=Size(substb);
substb:=SmallGeneratingSet(substb);
AQP:=Group(substb);
SetSize(AQP,B);
C:=[List(substb,x->PreImagesRepresentative(AQiso,x)),substb]; fi;
od;
AQ:=Group(C[1]);
SetIsFinite(AQ,true);
SetIsGroupOfAutomorphismsFiniteGroup(AQ,true);
SetSize(AQ,Size(AQP)); #AQP:=Group(C[2]); # ensure small gen set #SetSize(AQP,Size(AQ));
makeaqiso(); fi; fi;
lastperm:=AQiso; else
lastperm:=fail; fi;
i:=i+1;
od;
return AQ;
end);
# find characteristic subgroups by splitting into elementary abelian, # homogeneous layers
BindGlobal("AGSRModuleLayerSeries",function(g)
local s,l,r,i,j,sy,hom,p,pcgs;
s:=ShallowCopy(DerivedSeriesOfGroup(g));
r:=SolvableRadical(Last(s)); if Size(r)>1then# cannot be last, as solvable
Append(s,DerivedSeriesOfGroup(r)); fi;
i:=2; while i<=Length(s) do if HasAbelianFactorGroup(s[i-1],s[i]) then
p:=Factors(IndexNC(s[i-1],s[i]))[1]; # is it a single prime? if not IsPrimePowerInt(IndexNC(s[i-1],s[i])) then
hom:=NaturalHomomorphismByNormalSubgroupNC(s[i-1],s[i]);
sy:=SylowSystem(Image(hom));
l:=[sy[1]]; for j in [2..Length(sy)-1] do
Add(l,ClosureGroup(Last(l),sy[j]));
od;
l:=Reversed(l);
l:=List(l,x->PreImage(hom,x));
Info(InfoMorph,6,"insert prime @",i);
s:=Concatenation(s{[1..i-1]},l,s{[i..Length(s)]}); elif not HasElementaryAbelianFactorGroup(s[i-1],s[i]) then # not elementary abelian -- pth powers suffice as abelian
l:=s[i]; for j in GeneratorsOfGroup(s[i-1]) do
l:=ClosureGroup(l,j^p);
od;
Info(InfoMorph,6,"insert ppower @",i);
s:=Concatenation(s{[1..i-1]},[l],s{[i..Length(s)]}); else # make module
pcgs:=ModuloPcgs(s[i-1],s[i]);
l:=LinearActionLayer(g,pcgs);
l:=GModuleByMats(l,GF(p)); # check for characteristic submodules
r:=MTX.BasisRadical(l); if Length(r)=0then
r:=MTX.BasisSocle(l); if Length(r)=l.dimension then # semisimple -- use homogeneous
sy:=List(MTX.CollectedFactors(l),x->x[1]); if Length(sy)>1then
r:=Minimum(List(sy,x->x.dimension));
sy:=Filtered(sy,x->x.dimension=r);
r:=[]; for j in sy do
Append(r,MTX.Homomorphisms(j,l));
od;
r:=Concatenation(r);
r:=Filtered(TriangulizedMat(r),x->not IsZero(x)); else
r:=fail; fi; fi; fi;
if r=fail then
i:=i+1; else
l:=s[i]; for j in r do;
l:=ClosureGroup(l,PcElementByExponents(pcgs,j));
od; if Size(l)<Size(s[i-1]) and Size(l)>Size(s[i]) then
Info(InfoMorph,6,"insert module @",i);
s:=Concatenation(s{[1..i-1]},[l],s{[i..Length(s)]}); else
i:=i+1; fi; fi; fi; else
i:=i+1; fi;
od;
return s; end);
# find corresponding characteristic subgroups
BindGlobal("AGSRMatchedCharacteristics",function(g,h)
local a,props,cg,ch,clg,clh,ng,nh,coug,couh,pg,ph,i,j,stop,coinc;
props:=function(a)
local p,b;
if ID_AVAILABLE(Size(a))<>fail then
p:=ShallowCopy(-IdGroup(a)); # negative avoids clash with others else
p:=[Size(a)];
b:=ShallowCopy(AbelianInvariants(a));
Sort(b);
Add(p,b);
Add(p,List(DerivedSeriesOfGroup(a),Size)); fi; # # intersections # for i in [1..Length(chars)] do # for j in AGSRModuleLayerSeries(chars[i]) do # der(Intersection(j,a)); # der(ClosureGroup(j,a)); # od; # od;
return p; end;
# Process that G-subgroup i matches H-subgroup j. Return true if there is # a problem.
coinc:=function(i,j)
local a,b,sa,sb,sel,p,q; if pg[i]<>ph[j] then Error("matching bug");fi;
a:=cg[i];
b:=ch[j];
Add(ng,a);
Add(nh,b);
sel:=Difference([1..Length(cg)],[i]);
cg:=cg{sel};
pg:=pg{sel};
sel:=Difference([1..Length(ch)],[j]);
ch:=ch{sel};
ph:=ph{sel}; # find associated modules and further match
sa:=AGSRModuleLayerSeries(a);
sb:=AGSRModuleLayerSeries(b); if List(sa,Size)<>List(sb,Size) then return true;fi; for i in [1..Length(sa)] do
p:=Position(cg,sa[i]);
q:=Position(ch,sb[i]); if p=fail then if q<>fail then return true;fi; elif q=fail then return true; else if pg[p]<>ph[q] # but not same properties
or coinc(p,q) then return true;fi; fi;
od;
return false; end;
ng:=[];
nh:=[];
# get the characteristic subgroups
cg:=ShallowCopy(CharacteristicSubgroups(g));
ch:=ShallowCopy(CharacteristicSubgroups(h));
SortBy(cg,x->-Size(x));
SortBy(ch,x->-Size(x));
# find list of properties, tryign to match them up
pg:=List(cg,props);
ph:=List(ch,props);
stop:=false :=+1 while Length(cg)>0 and not; # test in loop as list changes if Collected(pg)<>Collected(ph) then return fail;fi;
i:=First([1..Length(pg)],x->Number(pg,y->y=pg[x])=1); if i<>fail then # found a unique one -- process # because properties agree this must match
j:= if coinc(i,j) then j:=etpg); else
stop:=true; # give up, for the moment # try clusters
j:=Set(pg);
clg=(j,x->cg{Filtered([1..Length(cg)],y->pg[y]=x)});
clh:=List(j,x->ch{(1.Length(),->y=)); # also use classes of size 1 for compare
Append(clg,List(ng,x->[x]));
Append(clh,List(nh, )java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 # sort larger first
SortParallel(clg,clh,function(a,b) return Size(a[1])>Size(b[1]);end);
i:=1; while i<Length(clg) do if Length(clg[i])>1then
j:=i+1; while j<=Length(clg) do
coug:=List(clg[i],x->Number(clg[j],y->IsSubset(x,y)));
couh:=List(clh[i],x->Number(clh[j],y->IsSubset(x,y))); if Collected(coug)<>Collected(couh) then return S:=];
a:=First(Collected(coug),x->x[2 i=1; if a>fail then # unique number -- split
a:=a[1]; if coinc(Position(cg,clg[i][Position(coug,a)]),
Position(ch,clh[i][Position(couh,a)])) then return fail;fi;
i:=Length(clg); j:=Length(clg); # break out of loops
stop:=false; fi;
j:=j+1;
od; fi;
i:=i+1;
od; fi;
od;
BindGlobal("AGBoundedOrbrep",function(G,from,to,act,bound)
local orb,rep,S,i,g,img,p;
orb:=[from];
rep:=[One(G)];
S:=[];
i:=1; while i<=Length(orb) do for g in GeneratorsOfGroup(G) do
img:=act(orb[i],g);
p:=Position(orb,img); if p=fail then if Length(orb)>=bound then return fail;fi; # length bailout
Add(orb,img);
Add(rep,rep[i]*g); else
img:=rep[i]*g/rep[p]; if not (img in S or img^-1 in S) then
Add(S,img); fi; fi;
od;
i:=i+1;
od;
p:=Position(orb,to); if p=fail then return false;fi; # not in orbit
return rec(rep:=rep[p],orblen:=Length(orb),stabgens:=S); end);
# pathetic isomorphism test, based on the automorphism group of GxH. This is # only of use as long as we don't yet have a Cannon/Holt version of # isomorphism available and there are many generators
InstallGlobalFunction(PatheticIsomorphism,function(G,H)
local d,a,map,cG,nG,nH,i,j,u,v,asAutomorphism,K,L,conj,e1,e2,
iso,api,gens,pre,aab,as,somechar;
# TODO: use matgrp package if not (IsPermGroup(G) or IsPcGroup(G)) then
i:=IsomorphismPermGroup(G);
iso:=PatheticIsomorphism(Image(i,G),H); if iso=fail then
return iso; else
return i*iso; fi; fi;
# TODO: use matgrp package if not (IsPermGroup(H) or IsPcGroup(H)) then
i:=IsomorphismPermGroup(H);
iso:=PatheticIsomorphism(G,Image(i,H)); if iso=fail then
return iso; else
return iso*InverseGeneralMapping(i); fi; fi;
# go through factors of characteristic series to keep orbits short.
AutomorphismGroup(G:someCharacteristics:=fail);
AutomorphismGroup(H:someCharacteristics:=fail);
d:=AGSRMatchedCharacteristics(G,H); if d=fail then return fail;fi; # characteristics do not match
nG:=d.ng;
nH:=d.nh; for i in [1..Length(d.cg)] do
u:=TrivialSubgroup(G); for j in d.cg[i] do
u:=ClosureGroup(u,j);
od; if not u in nG then
Add(nG,u);
u:=TrivialSubgroup(H); for j in d.ch[i] do
u:=ClosureGroup(u,j);
od;
Add(nH,u); fi;
od;
d:=DirectProduct(G,H);
e1:=Embedding(d,1);
e2:=Embedding(d,2); # combine images of characteristic factors, reverse order
cG:=[];
nG:=Reversed(nG);
nH:=Reversed(nH); for i in [1..Length(nG)] do
Add(cG,ClosureGroup(Image(e1,nG[i]),Image(e2,nH[i])));
od;
nG:=Concatenation([TrivialSubgroup(G)],nG);
nH:=Concatenation([TrivialSubgroup(H)],nH);
SortParallel(nG,nH,function(a,b) return Size(a)<Size(b);end); if List(nG,Size)<>List(nH,Size) then return fail;fi;
for i in [2..Length(nG)] do
K:=Filtered([1..Length(nG)],x->Size(nG[x])*2=Size(nG[i])
and IsSubset(nG[i],nG[x])); if Length(K)>0then
K:=K[1]; # We are seeking an isomorphism, not the full automorphism group of # GxG. It is thus sufficient, if we find the subgroup Aut(G)\wr 2.
# We now found that G and H have two characteristic subgroups A<B with # [B:A]=2. An isomorphism swapping G and H will need to map B to B and # A to A. Furthermore, in the factor modulo A_G xA_H, a generator of # B_G must be swappes with a generator of B_H. # This implies that A_G\times A_H, together with the diagonal of B is # characteristic in Aut(A)\wr 2. We thus may add this subgroup as # ``characteristic'' to improve the series.
Add(cG,ClosureGroup(
ClosureGroup(Image(e1,nG[K]),Image(e2,nH[K])),
Image(e1,First(GeneratorsOfGroup(nG[i]),x->not x in nG[K]))
*Image(e2,First(GeneratorsOfGroup(nH[i]),x->not x in nH[K]))));
fi;
od;
aab:=[Image(e1,G),Image(e2,H)]; # we also fix the *pairs* of the characteristic subgroups as orbits. Again # this must happen in Aut(G)\wr 2, and reduces the size of the group.
somechar:=rec(subgroups:=cG,
orbits:=List([1..Length(nG)],x->[Image(e1,nG[x]),Image(e2,nH[x])]));
a:=AutomorphismGroup(d:autactbase:=aab,someCharacteristics:=somechar,
directs:=aab,
delaypermrep:=true ); for i in cG do if not ForAll(GeneratorsOfGroup(a),x->Image(x,i)=i) then
a:=Stabilizer(a,i,asAutomorphism); fi;
od;
# now work in reverse through the characteristic factors
conj:=One(a);
K:=Image(e1,G);
L:=Image(e2,H);
map:=AGBoundedOrbrep(a,K,L,asAutomorphism,20); if map=false then
Info(InfoMorph,1,"Shortorb test noniso");
return fail; elif map<>fail then
conj:=map.rep;
Info(InfoMorph,1,"Shortorb test iso found"); else
as:=a;
Add(cG,TrivialSubgroup(d));
SortBy(cG,x->-Size(x)); for i in cG do
u:=ClosureGroup(i,K);
v:=ClosureGroup(i,L); if u<>v then
# try cheap orbit stabilizer first if iso<>fail then
map:=fail; else
map:=AGBoundedOrbrep(as,u,v,asAutomorphism,100); fi; if map=false then
Info(InfoMorph,1,"Shortorb factor noniso");
return fail; elif map<>fail then
Info(,,Shortorb factorreduce ",map.orblen);
as:=SubgroupNC(Parent(as),map.stabgens);
map:=map.rep;
conj:=conj*map;
K:=Image(map,K);
as:=as^map; else if iso=fail then
Info(InfoMorph,1,"Shortorb failed, get delayed permrep"); if IsBound(a!.makeaqiso) then a!.makeaqiso();fi;
iso:=IsomorphismPermGroup(a:autactbase:=aab);
api:=Image(iso,as); fi;
if IsSolvableGroup(api) then
gens:=Pcgs(api); else
gens:=mallGeneratingSet(api); fi;
pre:=List(gens,x->PreImagesRepresentative(iso,x));
map:=RepresentativeAction(SubgroupNC(a,pre),u,v,asAutomorphism); if map= then
return fail; fi;
conj:=conj*map;
K:=Image(map,K);
# isomorphism available and there are many generators
u:=Stabilizer(api,v,gens,pre,asAutomorphism);
Info(nfoMorph," ,Size(d)/Size(i),""
reduce ",Size(api)/Size())
api:=u; fi; fi; fi;
od;
fi;
return GroupHomomorphismByImagesNC(G,H,java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 47
List(GeneratorsOfGroup(G),x->;
Image(conj,Image(e1,x java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28 end);
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