Merge pull request #896 from hulpke/additions: Further enhancements t…

…o group automorphisms

This PR contains a number of improvements and additions for testing of group isomorphism, respectively calculation of automorphism groups. It has been tested over several months and is merged to avoid too much divergence between master and development branches.

The main effect of this change should be a significantly improved performance for group isomorphism/automorphisms.

The main changes are:

- The automorphism based isomorphism test is also used for solvable groups, unless they can be generated by few elements.
- To this end, the automorphism group calculation allows the (internal) specification of characteristic (or to-be-stabilized) subgroups.
- When calculating subspace stabilizers in GL, write down a generating set instead of performing an orbit/stabilizer calculation.
- Improvements of heuristics in the code which represents factor groups as permutation groups and determines smaller degree permutation representations, to avoid falling into certain long-runtime traps. Code may utilize maximal subgroups.
- A dedicated test file

lib/autsr.gi

@@ -1,6 +1,6 @@
 #############################################################################
 ##
-#W  auttf.gi                GAP library                      Alexander Hulpke
+#W  autsr.gi                GAP library                      Alexander Hulpke
 #W                                                           Soley Jonsdottir
 ##
 ##
@@ -149,7 +149,12 @@ 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,bas,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;
+      scharorb,asAutom,jorb,jorpo,substb,isBadPermrep,ma;
+
+  # criterion for when to force degree reduction
+  isBadPermrep:=function(g)
+    return NrMovedPoints(g)^2>Size(g)*Index(g,DerivedSubgroup(g));
+  end;
 
   asAutom:=function(sub,hom) return Image(hom,sub);end;
 
@@ -170,20 +175,23 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
     AQP:=Image(AQiso,AQ);
     # force degree down
     a:=Size(AQP);
-    AQP:=Group(SmallGeneratingSet(AQP));
+    AQP:=Group(SmallGeneratingSet(AQP),One(AQP));
     SetSize(AQP,a);
-    a:=SmallerDegreePermutationRepresentation(AQP:cheap);
-    if NrMovedPoints(Image(a))<NrMovedPoints(AQP) then
-      Info(InfoMorph,3,"Permdegree reduced ",
-	    NrMovedPoints(AQP),"->",NrMovedPoints(Image(a)));
-      AQiso:=AQiso*a;
-      b:=Image(a,AQP);
-      if Length(GeneratorsOfGroup(b))>Length(GeneratorsOfGroup(AQP)) then
-	b:=Group(List(GeneratorsOfGroup(AQP),x->ImagesRepresentative(a,x)));
-	SetSize(b,Size(AQP));
+    if isBadPermrep(AQP) then
+      a:=SmallerDegreePermutationRepresentation(AQP:cheap);
+      if NrMovedPoints(Image(a))<NrMovedPoints(AQP) then
+	Info(InfoMorph,3,"Permdegree reduced ",
+	      NrMovedPoints(AQP),"->",NrMovedPoints(Image(a)));
+	AQiso:=AQiso*a;
+	b:=Image(a,AQP);
+	if Length(GeneratorsOfGroup(b))>Length(GeneratorsOfGroup(AQP)) then
+	  b:=Group(List(GeneratorsOfGroup(AQP),x->ImagesRepresentative(a,x)));
+	  SetSize(b,Size(AQP));
+	fi;
+	AQP:=b;
       fi;
-      AQP:=b;
     fi;
+
   end;
 
   stablim:=function(gp,cond,lim)
@@ -313,6 +321,8 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
     Q:=Image(hom,G);
   fi; 
 
+  ma:=MaximalSubgroupClassesSol(G);
+
   AQ:=AutomorphismGroupFittingFree(Q:someCharacteristics:=fail);
   AQI:=InnerAutomorphismsAutomorphismGroup(AQ);
   lastperm:=fail;
@@ -329,19 +339,41 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
       hom:=NaturalHomomorphismByNormalSubgroup(G,N);
       Q:=Image(hom,G);
       # degree reduction called for?
-      if Size(N)>1 and IsPermGroup(Q) and NrMovedPoints(Q)^2>Size(Q) then
+      if Size(N)>1 and isBadPermrep(Q) then
+	#if NrMovedPoints(Q)>15000 then Error("egad!");fi;
 	q:=SmallerDegreePermutationRepresentation(Q);
 	Info(InfoMorph,3,"reduced permrep Q ",NrMovedPoints(Q)," -> ",
 	     NrMovedPoints(Range(q)));
 	hom:=hom*q;
 	Q:=Image(hom,G);
       fi;
+
+      # inherit radical factor map
+      q:=GroupHomomorphismByImagesNC(Q,Range(ff.factorhom),
+	List(GeneratorsOfGroup(G),x->ImagesRepresentative(hom,x)),
+	List(GeneratorsOfGroup(G),x->ImagesRepresentative(ff.factorhom,x)));
+      b:=Image(hom,ff.radical);
+      SetRadicalGroup(Q,b);
+      AddNaturalHomomorphismsPool(Q,b,q);
+
+      # Use known maximals for Frattini
+      for j in ma do
+        D:=Image(hom,j);
+	if not IsSubset(D,b) then
+	  b:=Core(Q,NormalIntersection(b,D));
+	fi;
+      od;
+      SetIsNilpotentGroup(b,true);
+      SetFrattiniSubgroup(Q,b);
+
+      # M-factor
       Mim:=Image(hom,M);
       MPcgs:=Pcgs(Mim);
       q:=GroupHomomorphismByImagesNC(Q,OQ,
 	List(GeneratorsOfGroup(G),x->ImagesRepresentative(hom,x)),
 	List(GeneratorsOfGroup(G),x->ImagesRepresentative(lhom,x)));
       AddNaturalHomomorphismsPool(Q,Mim,q);
+
       mo:=GModuleByMats(LinearActionLayer(GeneratorsOfGroup(Q),MPcgs),GF(RelativeOrders(MPcgs)[1]));
       # is the extension split?
       ocr:=OneCocycles(Q,Mim);
@@ -543,7 +575,7 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
     # desperately try to grab some further generators
     #stablim(sub,cond,10000)=false then
 
-    #if Size(sub)/Size(Aperm)>100000 then Error("HundredK"); fi;
+    #if Size(sub)/Size(Aperm)>1000000 then Error("Million"); fi;
     sub:=SubgroupProperty(sub,cond,Aperm);
 
     Aperm:=Group(Apa,());
@@ -597,19 +629,34 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
      then
 
       if rada=fail then
-	ind:=IsomorphismPcGroup(r);
-	rada:=AutomorphismGroup(Image(ind,r):someCharacteristics:=fail,actbase:=fail);
-	# we only consider those homomorphism that stabilize the series we use
-	for k in List(ser,x->Image(ind,x)) do
-	  if ForAny(GeneratorsOfGroup(rada),x->Image(x,k)<>k) then
-	    Info(InfoMorph,3,"radical automorphism stabilizer");
-	    NiceMonomorphism(rada:autactbase:=fail,someCharacteristics:=fail);
-	    rada:=Stabilizer(rada,k,asAutom);
-	  fi;
-	od;
-	# move back to bad degree
-	rada:=Group(List(GeneratorsOfGroup(rada),
-	  x-> InducedAutomorphism(InverseGeneralMapping(ind),x)));
+	if IsElementaryAbelian(r) and Size(r)>1 then
+	  B:=Pcgs(r);
+	  rf:=GF(RelativeOrders(B)[1]);
+	  ind:=Filtered(ser,x->IsSubset(r,x) and Size(x)>1 and Size(x)<Size(r)); 
+	  ind:=List(ind,x->List(GeneratorsOfGroup(x),y->ExponentsOfPcElement(B,y)));
+	  ind:=List(ind,x->x*One(rf));
+	  ind:=SpaceAndOrbitStabilizer(Length(B),rf,ind,[]);
+	  rada:=List(GeneratorsOfGroup(ind),x->
+	    GroupHomomorphismByImagesNC(r,r,B,List(x,y->PcElementByExponents(B,List(y,Int)))));
+	  rada:=Group(rada);
+	  SetIsGroupOfAutomorphismsFiniteGroup(rada,true);
+	  NiceMonomorphism(rada:autactbase:=fail,someCharacteristics:=fail);
+	else
+	  ind:=IsomorphismPcGroup(r);
+	  rada:=AutomorphismGroup(Image(ind,r):someCharacteristics:=fail,actbase:=fail);
+	  # we only consider those homomorphism that stabilize the series we use
+	  for k in List(ser,x->Image(ind,x)) do
+	    if ForAny(GeneratorsOfGroup(rada),x->Image(x,k)<>k) then
+	      Info(InfoMorph,3,"radical automorphism stabilizer");
+	      NiceMonomorphism(rada:autactbase:=fail,someCharacteristics:=fail);
+	      rada:=Stabilizer(rada,k,asAutom);
+	    fi;
+	  od;
+	  # move back to bad degree
+	  rada:=Group(List(GeneratorsOfGroup(rada),
+	    x-> InducedAutomorphism(InverseGeneralMapping(ind),x)));
+
+	fi;
       fi;
 
       rf:=Image(hom,r);
@@ -674,18 +721,25 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
 
 	if Length(u)>0 then
 	  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:=Orbit(AQP,j[1],C[2],C[1],asAutom);
+	      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:=PreImage(B,Stabilizer(Image(B),Set(jorpo),OnSets));
+		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
@@ -753,6 +807,8 @@ local d,a,map,possibly,cG,cH,nG,nH,i,j,sel,u,v,asAutomorphism,K,L,conj,e1,e2,
   end;
 
   # go through factors of characteristic series to keep orbits short.
+  AutomorphismGroup(G:someCharacteristics:=fail);
+  AutomorphismGroup(H:someCharacteristics:=fail);
   cG:=CharacteristicSubgroupsLib(G);
   nG:=[];
   cH:=ShallowCopy(CharacteristicSubgroupsLib(H));
@@ -780,7 +836,7 @@ local d,a,map,possibly,cG,cH,nG,nH,i,j,sel,u,v,asAutomorphism,K,L,conj,e1,e2,
                                  x->possibly(cH[i],cH[x])));
 	v:=TrivialSubgroup(H);
 	for j in sel do
-	  u:=ClosureGroup(v,cH[j]);
+	  v:=ClosureGroup(v,cH[j]);
 	od;
 	if Size(u)<>Size(v) then
 	  return fail;

lib/factgrp.gi

@@ -314,10 +314,32 @@ local G,pool,p,comb,i,c,perm,l,isi,N,discard,ab;
   if Length(arg)>1 then
     N:=arg[2];
     p:=Filtered(p,i->IsSubset(pool.ker[i],N));
+    if Length(p)=0 then
+      return;
+    fi;
+    SortParallel(List(pool.ker{p},Size),p);
+    if Size(pool.ker[p[1]])=Size(N) and Length(p)>3 then
+      # N in pool
+      c:=pool.cost[p[1]];
+      p:=Filtered(p,x->pool.cost[x]<c);
+    fi;
   else
     N:=fail;
   fi;
 
+  # minimal ones
+  c:=Filtered([1..Length(p)],
+      x->not ForAny([1..x-1],y->IsSubset(pool.ker[p[x]],pool.ker[p[y]])));
+  c:=p{c};
+  if Size(Intersection(pool.ker{c}))>Size(N) then
+    # cannot reach N
+    return; 
+  elif Length(p)>20 or Size(N)=1 then
+    p:=c; # use only minimal ones if there is lots
+  fi;
+
+  #if Length(p)>20 then Error("hier!");fi;
+
   # do the abelians extra.
   p:=Filtered(p,x->not HasAbelianFactorGroup(G,pool.ker[x]));
 
@@ -523,7 +545,7 @@ end);
 BADINDEX:=1000; # the index that is too big
 GenericFindActionKernel:=function(arg)
 local G, N, knowi, goodi, simple, uc, zen, cnt, pool, ise, v, bv, badi,
-totalcnt, interupt, u, nu, cor, zzz,bigperm,perm,badcores;
+totalcnt, interupt, u, nu, cor, zzz,bigperm,perm,badcores,max,i;
 
   G:=arg[1];
   N:=arg[2];
@@ -635,7 +657,7 @@ totalcnt, interupt, u, nu, cor, zzz,bigperm,perm,badcores;
 	fi;
 	totalcnt:=totalcnt+1;
 	if KnownNaturalHomomorphismsPool(G,N) and
-	  Minimum(Index(G,v),knowi)<20000 
+	  Minimum(Index(G,v),knowi)<100000 
 	     and 5*totalcnt>Minimum(Index(G,v),knowi,1000) then
 	  # interupt if we're already quite good
 	  interupt:=true;
@@ -673,7 +695,7 @@ totalcnt, interupt, u, nu, cor, zzz,bigperm,perm,badcores;
       if Size(cor)>Size(N) and IsSubset(cor,N) and not cor in badcores then
 	Add(badcores,cor);
       fi;
-      # store known information(we do't act, just store the subgroup.
+      # store known information(we do't act, just store the subgroup).
       # Thus this is fairly cheap
       pool.dotriv:=true;
       zzz:=AddNaturalHomomorphismsPool(G,cor,u,Index(G,u));
@@ -686,6 +708,17 @@ totalcnt, interupt, u, nu, cor, zzz,bigperm,perm,badcores;
       zzz:=DegreeNaturalHomomorphismsPool(G,N);
 
       Info(InfoFactor,3,"  ext ",cnt,": ",Index(G,u)," best degree:",zzz);
+      if Size(cor)>Size(N) and Index(G,u)*2<knowi and
+      ValueOption("inmax")=fail then
+        max:=Filtered(MaximalSubgroupClassReps(u:inmax,cheap),
+	  x->IndexNC(G,x)<knowi and IsSubset(x,N)); 
+        for i in max do
+	  cor:=Core(G,i);
+	  AddNaturalHomomorphismsPool(G,cor,i,Index(G,i));
+	od;
+	zzz:=DegreeNaturalHomomorphismsPool(G,N);
+	Info(InfoFactor,3,"  Maxes: ",Length(max)," best degree:",zzz);
+      fi;
     else
       zzz:=DegreeNaturalHomomorphismsPool(G,N);
     fi;

lib/grpffmat.gi

@@ -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,
@@ -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 );
 
 

lib/grplatt.gi

@@ -1091,7 +1091,12 @@ InstallGlobalFunction(LatticeViaRadical,function(arg)
 
 	  if pcgs=false then
 	    lmpc:=ModuloPcgs(ser[i-1],nts[j]);
-	    npcgs:=ModuloPcgs(nts[j],ser[i]);
+	    if Size(nts[j])=1 and Size(ser[i])=1 then
+	      # avoid degenerate case
+	      npcgs:=Pcgs(nts[j]);
+	    else
+	      npcgs:=ModuloPcgs(nts[j],ser[i]);
+	    fi;
 	  else
 	    if IsTrivial(nts[j]) then
 	      lmpc:=pcgs[i-1];
@@ -2321,7 +2326,7 @@ local rt,op,a,l,i,j,u,max,subs;
   return rec(subgroups:=List(a,i->ClosureGroup(U,rt{i})),inclusions:=max);
 end);
 
-InstallMethod(IntermediateSubgroups,"blocks for coset operation",
+InstallMethod(IntermediateSubgroups,"using maximal subgroups",
   IsIdenticalObj, [IsGroup,IsGroup],
   1, # better than previous if index larger
 function(G,U)

lib/grppc.gi

@@ -592,6 +592,7 @@ InstallMethod( SubgroupByPcgs, "subgroup with pcgs",
 function( G, pcgs )
     local U;
     U := SubgroupNC( G, AsList( pcgs ) );
+    SetSize(U,Product(RelativeOrders(pcgs)));
     SetPcgs( U, pcgs );
     SetGroupOfPcgs (pcgs, U);
     # home pcgs will be inherited

lib/grppcaut.gd

@@ -7,12 +7,15 @@
 #Y  Copyright (C) 2002 The GAP Group
 ##
 
+DeclareGlobalFunction("SpaceAndOrbitStabilizer");
+
 #############################################################################
 ##
 #P IsFrattiniFree
 ##
 DeclareProperty( "IsFrattiniFree", IsGroup );
 
+
 DeclareGlobalFunction("AutomorphismGroupNilpotentGroup");
 DeclareGlobalFunction("AutomorphismGroupSolvableGroup");
 DeclareGlobalFunction("AutomorphismGroupFrattFreeGroup");