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ENHANCE: Improved subspace stabilizer in case of full GL.
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@hulpke
hulpke committed Oct 10, 2016
1 parent 97b0e6a commit 2bf6b36
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20 changes: 12 additions & 8 deletions lib/grpffmat.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -90,8 +90,10 @@ InstallMethod( IsNaturalGL,
0,

function( grp )
return Size( grp ) = Size( GL( DimensionOfMatrixGroup( grp ),
Size( FieldOfMatrixGroup( grp ) ) ) );
return MTX.IsAbsolutelyIrreducible(
GModuleByMats(GeneratorsOfGroup(grp),DefaultFieldOfMatrixGroup(grp))) and
Size( grp ) = Size( GL( DimensionOfMatrixGroup( grp ),
Size( FieldOfMatrixGroup( grp ) ) ) );
end );

InstallMethod( IsNaturalSL,
Expand All @@ -101,12 +103,14 @@ InstallMethod( IsNaturalSL,
0,

function( grp )
local gen, d, f;
f := FieldOfMatrixGroup( grp );
d := DimensionOfMatrixGroup( grp );
gen := GeneratorsOfGroup( grp );
return ForAll(gen, x-> DeterminantMat(x) = One(f))
and Size(grp) = Size(SL(d, Size(f)));
local gen, d, f;
f := FieldOfMatrixGroup( grp );
d := DimensionOfMatrixGroup( grp );
gen := GeneratorsOfGroup( grp );
return MTX.IsAbsolutelyIrreducible(
GModuleByMats(GeneratorsOfGroup(grp),DefaultFieldOfMatrixGroup(grp))) and
ForAll(gen, x-> DeterminantMat(x) = One(f))
and Size(grp) = Size(SL(d, Size(f)));
end );


Expand Down
194 changes: 131 additions & 63 deletions lib/grppcaut.gi
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Original file line number Diff line number Diff line change
Expand Up @@ -897,7 +897,7 @@ end;
#F AutomorphismGroupSolvableGroup( G )
##

# shoul db emore generic -- test whether orbit length exceeds limit
# should be more generic -- test whether orbit length exceeds limit
BindGlobal("OrbitIsLonger",function(acts,pnt,act,lim)
local orb,d,gen,i,p,D;
# try to find a domain
Expand All @@ -922,6 +922,64 @@ local orb,d,gen,i,p,D;
return false;
end);

# another function that should be more general and has space for improvement.
BindGlobal("SubspaceStabilizerMatrixGroup",function(G,bas)
local f,act,xset,stb,hom,n,m,l,i;
n:=DimensionOfMatrixGroup(G);
if Length(bas)=0 or ForAll(GeneratorsOfGroup(G),
x->ForAll(bas,y->SolutionMat(bas,y*x)<>fail)) then
# space is fixed
return G;
fi;
bas:=OnSubspacesByCanonicalBasis(bas,One(G)); # canonical form
f:=DefaultFieldOfMatrixGroup(G);
if IsNaturalGL(G) then
# write down stabilizer
l:=Length(bas);
act:=BaseSteinitzVectors(IdentityMat(n,f),
bas);
act:=Concatenation(bas,act.factorspace);
stb:=[];
# subspace
for i in GeneratorsOfGroup(GL(l,f)) do
m:=IdentityMat(n,f);
m{[1..l]}{[1..l]}:=i;
Add(stb,m);
od;
# factor
for i in GeneratorsOfGroup(GL(n-l,f)) do
m:=IdentityMat(n,f);
m{[l+1..n]}{[l+1..n]}:=i;
Add(stb,m);
od;
# mixer
m:=IdentityMat(n,f);
m[l+1][1]:=One(f);
Add(stb,m);
stb:=List(stb,x->x^act); # base change
stb:=Group(stb);
SetSize(stb,Size(GL(l,f))*Size(GL(n-l,f))*Size(f)^(l*(n-l)));
Info(InfoMatOrb,2,"reduced full GL by ",Size(G)/Size(stb));
return stb;
fi;
if not OrbitIsLonger(GeneratorsOfGroup(G),bas,OnSubspacesByCanonicalBasis,
Size(f)^QuoInt(Length(bas[1]),2)) then
#just plain
act:=OnSubspacesByCanonicalBasis;
xset:=Orbit(G,bas,act);
stb:=[Position(xset,bas)];
else
act:=OnLines;
stb:=NormedVectors(VectorSpace(f,bas));
xset:=Union(Orbits(G,stb,act));
stb:=Set(List(stb,x->Position(xset,x)));
fi;
hom:=ActionHomomorphism(G,xset,act,"surjective");
act:=PreImage(hom,Stabilizer(Image(hom),stb,OnSets));
Info(InfoMatOrb,2,"reduce ",Size(G), " to ",Size(act));
return act;
end);

InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
local spec, weights, first, m, pcgsU, F, pcgsF, A, i, s, n, p, H,
pcgsH, pcgsN, N, epi, mats, M, autos, ocr, elms, e, list, imgs,
Expand Down Expand Up @@ -964,46 +1022,46 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
B := NormalizingReducedGL( spec, 1, first[2], M );

if somechar<>fail then
field:=DefaultFieldOfMatrixGroup(B);
C:=List(somechar,x->quotimg(F,FamilyPcgs(F),x));
C:=List(C,x->[x]);
C:=List(C,x->List(SmallGeneratingSet(x),
x->ExponentsOfPcElement(FamilyPcgs(F),x)*One(field)));
C:=Filtered(C,x->Length(x)>0);
C:=List(C,x->Filtered(OnSubspacesByCanonicalBasis(x,One(B)),
y->not IsZero(y)));
C:=Unique(C);
for D in C do
B:=SubspaceStabilizerMatrixGroup(B,D);
od;
if scharorb<>fail then
Append(C,List(scharorb,x->List(x,x->quotimg(F,FamilyPcgs(F),x))));
C:=List(scharorb,x->List(x,x->quotimg(F,FamilyPcgs(F),x)));
C:=Filtered(C,x->Size(x[1])>1 and Size(x[1])<Size(F));
C:=List(C,Set);
D:=Unique(C);
for C in D do
C:=List(C,x->List(SmallGeneratingSet(x),
x->ExponentsOfPcElement(FamilyPcgs(F),x)*One(field)));
C:=List(C,x->OnSubspacesByCanonicalBasis(x,One(B)));

if Length(C)=1 then
B:=SubspaceStabilizerMatrixGroup(B,C[1]);
else
act:=OnSubspacesByCanonicalBasis;
xset:=Orbit(B,C[1],act);
C:=Filtered(C,x->x in xset);
if Length(xset)>Length(C) then
hom:=ActionHomomorphism(B,xset,act,"surjective");
C:=List(C,x->Position(xset,x));
e:=Size(B);
B:=PreImage(hom,Stabilizer(Image(hom),C,OnSets));
Info(InfoAutGrp,2,"characteristics reduce ",e,
" to ",Size(B));
fi;
fi;
od;
fi;
C:=Filtered(C,x->Size(x[1])>1 and Size(x[1])<Size(F));
C:=List(C,Set);
D:=Unique(C);
field:=DefaultFieldOfMatrixGroup(B);
for C in D do
C:=List(C,x->List(SmallGeneratingSet(x),
x->ExponentsOfPcElement(FamilyPcgs(F),x)*One(field)));
C:=List(C,x->OnSubspacesByCanonicalBasis(x,One(B)));

if Length(C)=1 and
ForAny(GeneratorsOfGroup(B),
x->ForAny(C[1],y->SolutionMat(C[1],y*x)=fail))
and OrbitIsLonger(GeneratorsOfGroup(B),
OnSubspacesByCanonicalBasis(C[1],One(B)),
OnSubspacesByCanonicalBasis,Size(F)*2/3)
then
C:=NormedVectors(VectorSpace(field,C[1]));
act:=OnLines;
xset:=Union(Orbits(B,C,act));
else
act:=OnSubspacesByCanonicalBasis;
xset:=Orbit(B,C[1],act);
C:=Filtered(C,x->x in xset);
fi;
if Length(xset)>Length(C) then
hom:=ActionHomomorphism(B,xset,act,"surjective");
C:=List(C,x->Position(xset,x));

e:=Size(B);
B:=PreImage(hom,Stabilizer(Image(hom),C,OnSets));
Info(InfoAutGrp,2,"characteristics reduce ",e,
" to ",Size(B));
fi;
od;
fi;

A := AutomorphismGroupElAbGroup( F, B );
SetIsGroupOfAutomorphismsFiniteGroup(A,true);

Expand Down Expand Up @@ -1049,36 +1107,46 @@ InstallGlobalFunction(AutomorphismGroupSolvableGroup,function( G )
B:=SubgroupNC(B,SmallGeneratingSet(B));

if somechar<>fail then
field:=GF(RelativeOrders(pcgsN)[1]);
e:=Product(RelativeOrders(pcgsN));
D:=List(somechar,x->quotimg(H,pcgsH,Intersection(x,eN)));
D:=List(D,x->[x]);
# do something ...

if scharorb<>fail then
Append(D,List(scharorb,
x->List(x,x->quotimg(H,pcgsH,Intersection(x,eN)))));
fi;
C:=List(somechar,x->quotimg(H,pcgsH,Intersection(x,eN)));

D:=Filtered(D,x->Size(x[1])>1 and Size(x[1])<e);
D:=Unique(List(D,Set));

for C in D do
C:=List(C,x->List(SmallGeneratingSet(x),
C:=List(C,x->List(SmallGeneratingSet(x),
x->ExponentsOfPcElement(pcgsH,x){[s..n-1]}*One(field)));
C:=List(C,x->OnSubspacesByCanonicalBasis(x,One(B)));
xset:=Orbit(B,C[1],OnSubspacesByCanonicalBasis);
C:=Filtered(C,x->x in xset);
if Length(xset)>Length(C) then
hom:=ActionHomomorphism(B,xset,OnSubspacesByCanonicalBasis,
"surjective");
C:=List(C,x->Position(xset,x));

e:=Size(B);
B:=PreImage(hom,Stabilizer(Image(hom),C,OnSets));
Info(InfoAutGrp,2,"characteristics reduce ",e,
" to ",Size(B));
fi;
C:=Filtered(C,x->Length(x)>0);
C:=List(C,x->Filtered(OnSubspacesByCanonicalBasis(x,One(B)),
y->not IsZero(y)));
C:=Unique(C);
for D in C do
B:=SubspaceStabilizerMatrixGroup(B,D);
od;
if scharorb<>fail then
C:=List(scharorb,
x->List(x,x->quotimg(H,pcgsH,Intersection(x,eN))));
C:=Filtered(C,x->Size(x[1])>1 and Size(x[1])<Size(F));
D:=Unique(List(C,Set));
for C in D do
C:=List(C,x->List(SmallGeneratingSet(x),
x->ExponentsOfPcElement(pcgsH,x){[s..n-1]}*One(field)));
C:=List(C,x->OnSubspacesByCanonicalBasis(x,One(B)));

if Length(C)=1 then
B:=SubspaceStabilizerMatrixGroup(B,C[1]);
else
act:=OnSubspacesByCanonicalBasis;
xset:=Orbit(B,C[1],act);
C:=Filtered(C,x->x in xset);
if Length(xset)>Length(C) then
hom:=ActionHomomorphism(B,xset,act,"surjective");
C:=List(C,x->Position(xset,x));
e:=Size(B);
B:=PreImage(hom,Stabilizer(Image(hom),C,OnSets));
Info(InfoAutGrp,2,"characteristics reduce ",e,
" to ",Size(B));
fi;
fi;
od;
fi;
fi;

if weights[s][2] = 1 then
Expand Down
9 changes: 9 additions & 0 deletions tst/teststandard/grpauto.tst
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Original file line number Diff line number Diff line change
Expand Up @@ -87,6 +87,15 @@ fail
gap> IsomorphismGroups(G,PcGroupCode(CodePcGroup(G),Size(G)))=fail;
false

gap> G:=PcGroupCode(18738408935379049727906755356708168311565445686261463850856,
> 21952);;
gap> H:=PcGroupCode(18738408935359210460657231881739776911013615108923450662760,
> 21952);;
gap> IsomorphismGroups(G,H);
fail
gap> IsomorphismGroups(G,PcGroupCode(CodePcGroup(G),Size(G)))=fail;
false

#
# Too hard work for permiso
#
Expand Down

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