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ENHANCE: Performance improvements to PatheticIsomorphism
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Use characteristic factors to first map in factor groups (shorter orbits)
Use direct product structure to obtain better permrep for automorphism group.

FIX: Do not drag autactbase into solvable case.
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@hulpke
hulpke committed Aug 19, 2016
1 parent fd3fa45 commit aa5792d
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143 changes: 129 additions & 14 deletions lib/autsr.gi
View file
Original file line number Diff line number Diff line change
Expand Up @@ -145,11 +145,21 @@ BindGlobal("AutomGrpSR",function(G)
local ff,r,d,ser,u,v,i,j,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
b,fratsim,AQ,OQ,Zm,D,innC,bas,oneC,imgs,C,maut,innB,tmpAut,imM,a,A,B,
cond,sub,AQI,AQP,AQiso,rf,res,resperm,proj,
comiso,extra,mo,rada,makeaqiso,ind,lastperm;
comiso,extra,mo,rada,makeaqiso,ind,lastperm,actbase;

actbase:=ValueOption("autactbase");

makeaqiso:=function();
if HasIsomorphismPermGroup(AQ) then
AQiso:=IsomorphismPermGroup(AQ);
elif HasNiceMonomorphism(AQ) and IsPermGroup(Range(NiceMonomorphism(AQ))) then
AQiso:=NiceMonomorphism(AQ);
elif actbase<>fail then
AQiso:=IsomorphismPermGroup(AQ:autactbase:=List(actbase,x->Image(hom,x)));
else
AQiso:=IsomorphismPermGroup(AQ);
fi;
Info(InfoMorph,3,"Permrep of AQ ",Size(AQ));
AQiso:=IsomorphismPermGroup(AQ);
AQP:=Image(AQiso,AQ);
# force degree down
a:=Size(AQP);
Expand Down Expand Up @@ -424,15 +434,21 @@ local ff,r,d,ser,u,v,i,j,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
Append(gens,B);
Append(gens,A);

for j in gens do
SetIsBijective(j,true);
od;
A:=Group(gens);
SetIsAutomorphismGroup(A,true);
SetIsGroupOfAutomorphismsFiniteGroup(A,true);
SetIsFinite(A,true);

AQI:=SubgroupNC(A,innB);
SetInnerAutomorphismsAutomorphismGroup(A,AQI);

AQ:=A;
# use the actbase for order computations
if actbase<>fail then
Size(A:autactbase:=List(actbase,x->Image(hom,x)));
fi;

# do we use induced radical automorphisms to help next step?
if Size(KernelOfMultiplicativeGeneralMapping(hom))>1 and
Expand Down Expand Up @@ -466,7 +482,7 @@ local ff,r,d,ser,u,v,i,j,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,

if Size(ind)*100<Size(res) then
# reduce to subgroup that induces valid automorphisms
Info(InfoMorph,1,"Reduce by factor ",Size(res)/Size(ind));
Info(InfoMorph,1,"Radical autos reduce by factor ",Size(res)/Size(ind));
resperm:=IsomorphismPermGroup(res);
proj:=GroupHomomorphismByImagesNC(AQP,Image(resperm),
B[2],List(C,x->ImagesRepresentative(resperm,x)));
Expand Down Expand Up @@ -495,17 +511,116 @@ end);
# only of use as long as we don't yet have a Cannon/Holt version of
# isomorphism available and there are many generators
BindGlobal("PatheticIsomorphism",function(G,H)
local d,a,map;
local d,a,map,possibly,cG,cH,nG,nH,i,j,sel,u,v,asAutomorphism,K,L,conj,e1,e2,
iso,api,good,gens,pre;
possibly:=function(a,b)
if Size(a)<>Size(b) then
return false;
fi;
if AbelianInvariants(a)<>AbelianInvariants(b) then
return false;
fi;
if Size(a)<1000 and Size(a)<>512 and IdGroup(a)<>IdGroup(b) then
return false;
fi;
return true;
end;

asAutomorphism:=function(sub,hom)
return Image(hom,sub);
end;

# go through factors of characteristic series to keep orbits short.
cG:=CharacteristicSubgroupsLib(G);
nG:=[];
cH:=ShallowCopy(CharacteristicSubgroupsLib(H));
if Length(cG)<>Length(cH) then
return fail;
fi;
SortBy(cH,Size);
nH:=[];
i:=1;
good:=[1..Length(cH)];
while i<=Length(cH) do
if i in good and Size(cH[i])>1 and Size(cH[i])<Size(H) then
sel:=Filtered([1..Length(cG)],x->possibly(cG[x],cH[i]));
if Length(sel)=0 then
return fail;
elif Length(sel)=1 then
Add(nG,cG[sel[1]]);
Add(nH,cH[i]);
else
u:=TrivialSubgroup(G);
for j in sel do
u:=ClosureGroup(u,cG[j]);
od;
sel:=Concatenation([i],Filtered([i+1..Length(cH)],
x->possibly(cH[i],cH[x])));
v:=TrivialSubgroup(H);
for j in sel do
u:=ClosureGroup(v,cH[j]);
od;
if Size(u)<>Size(v) then
return fail;
fi;
good:=Difference(good,sel);
if Size(u)<Size(G) then
Add(nG,u);
Add(nH,v);
fi;
fi;
fi;
i:=i+1;
od;

d:=DirectProduct(G,H);
a:=AutomorphismGroup(d);
map:=RepresentativeAction(a,Image(Embedding(d,1)),
Image(Embedding(d,2)),
function(sub,hom)
return Image(hom,sub);
end);
if map=fail then return fail;fi;
e1:=Embedding(d,1);
e2:=Embedding(d,2);
# combine images of characteristic factors, reverse order
cG:=[];
for i in [Length(nG),Length(nG)-1..1] do
Add(cG,ClosureGroup(Image(e1,nG[i]),Image(e2,nH[i])));
od;

K:=[Image(e1,G),Image(e2,H)];
a:=AutomorphismGroup(d:autactbase:=K);
iso:=IsomorphismPermGroup(a:autactbase:=K);
api:=Image(iso);
#if NrMovedPoints(api)>5000 then
# K:=SmallerDegreePermutationRepresentation(api);
# Info(InfoMorph,2,"Permdegree reduced ",
# NrMovedPoints(api),"->",NrMovedPoints(Image(K)));
# iso:=iso*K;
# api:=Image(iso);
# fi;

# now work in reverse through the characteristic factors
conj:=One(a);
K:=Image(e1,G);
L:=Image(e2,H);
Add(cG,TrivialSubgroup(d));
for i in cG do
u:=ClosureGroup(i,K);
v:=ClosureGroup(i,L);
if u<>v then
gens:=SmallGeneratingSet(api);
pre:=List(gens,x->PreImagesRepresentative(iso,x));
map:=RepresentativeAction(SubgroupNC(a,pre),u,v,asAutomorphism);
if map=fail then
return fail;
fi;
conj:=conj*map;
K:=Image(map,K);

u:=Stabilizer(api,v,gens,pre,asAutomorphism);
Info(InfoMorph,1,"Factor ",Size(d)/Size(i),": ",
"reduce by ",Size(api)/Size(u));
api:=u;
fi;
od;

return GroupHomomorphismByImagesNC(G,H,GeneratorsOfGroup(G),
List(GeneratorsOfGroup(G),x->PreImagesRepresentative(Embedding(d,2),
Image(map,Image(Embedding(d,1),x)))));
List(GeneratorsOfGroup(G),x->PreImagesRepresentative(e2,
Image(conj,Image(e1,x)))));
end);

13 changes: 10 additions & 3 deletions lib/morpheus.gi
View file
Original file line number Diff line number Diff line change
Expand Up @@ -234,7 +234,8 @@ end);
##
# try to find a small faithful action for an automorphism group
InstallGlobalFunction(AssignNiceMonomorphismAutomorphismGroup,function(au,g)
local hom, allinner, gens, c, ran, r, cen, img, dom, u, subs, orbs, cnt, br, bv, v, val, o, i, comb, best;
local hom, allinner, gens, c, ran, r, cen, img, dom, u, subs, orbs, cnt, br, bv,
v, val, o, i, comb, best,actbase;

hom:=fail;
allinner:=HasIsAutomorphismGroup(au) and IsAutomorphismGroup(au);
Expand Down Expand Up @@ -378,10 +379,14 @@ local hom, allinner, gens, c, ran, r, cen, img, dom, u, subs, orbs, cnt, br, bv,
fi;
od;
else
actbase:=ValueOption("autactbase");
if actbase=fail then
actbase:=[g];
fi;
repeat
cnt:=cnt+1;
repeat
r:=Random(g);
r:=Random(Random(actbase));
until not r in u;
# force prime power order
if not IsPrimePowerInt(Order(r)) then
Expand Down Expand Up @@ -1924,7 +1929,9 @@ local A;
if HasIsFrattiniFree(G) and IsFrattiniFree(G) then
A:=AutomorphismGroupFrattFreeGroup(G);
else
A:=AutomorphismGroupSolvableGroup(G);
# currently autactbase does not work well, as the representation might
# change.
A:=AutomorphismGroupSolvableGroup(G:autactbase:=fail);
fi;
elif Size(RadicalGroup(G))=1 and IsPermGroup(G) then
# essentially a normalizer when suitably embedded
Expand Down

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