ENHANCE: Automorphism group computations

Reduce (or be more aggressive in reducing) number of generators and permutation degree for groups of automorphisms. Also, the cost of testing for class duplicates exceeds the gain of not testing duplicates.
gap-system · Oct 31, 2020 · 6a42bbf · 6a42bbf
1 parent 93abbf7
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diff --git a/lib/autsr.gi b/lib/autsr.gi
@@ -13,6 +13,73 @@
 ##    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
+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)>3 then
+      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
@@ -499,10 +566,16 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
     fi;
     AQP:=Image(AQiso,AQ);
     Info(InfoMorph,3,"Permrep of AQ ",Size(AQ),", deg:",NrMovedPoints(AQP));
+
+    if Length(GeneratorsOfGroup(AQP))=Length(GeneratorsOfGroup(AQ)) then
+      a:=AGSRReducedGens(AQP,AQP,AQ,AQiso);
+      if a<>fail then
+        AQP:=a[1];
+        AQ:=a[2];
+      fi;
+    fi;
+
     # force degree down
-    a:=Size(AQP);
-    AQP:=Group(SmallGeneratingSet(AQP),One(AQP));
-    SetSize(AQP,a);
     if isBadPermrep(AQP) then
       a:=SmallerDegreePermutationRepresentation(AQP:cheap);
       if NrMovedPoints(Image(a))<NrMovedPoints(AQP) then
@@ -705,8 +778,13 @@ 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);
       # degree reduction called for?
-      if Size(N)>1 and isBadPermrep(Q) then
+      if IsPermGroup(Q) and Size(N)>1 
+        and (isBadPermrep(Q) or NrMovedPoints(Q)>1000) then
         q:=SmallerDegreePermutationRepresentation(Q:cheap);
+        if NrMovedPoints(Q)>1000 
+          and NrMovedPoints(Q)=NrMovedPoints(Image(q)) then
+          q:=SmallerDegreePermutationRepresentation(Q);
+        fi;
         if NrMovedPoints(Range(q))<NrMovedPoints(Q) then
           Info(InfoMorph,3,"reduced permrep Q ",NrMovedPoints(Q)," -> ",
               NrMovedPoints(Range(q)));
@@ -948,23 +1026,34 @@ local ff,r,d,ser,u,v,i,j,k,p,bd,e,gens,lhom,M,N,hom,Q,Mim,q,ocr,split,MPcgs,
     sub:=SubgroupConditionAbove(sub,cond,Aperm);
     Info(InfoMorph,2,"end search ",j/Size(sub));
 
-    Aperm:=Group(Apa,());
+    # Note: Aperm is larger than what is generated by Apa
     j:=1;
     while Size(Aperm)<Size(sub) do
       ac:=InnerAutomorphism(OQ,Image(q,GeneratorsOfGroup(Q)[j]));
       k:=ImagesRepresentative(AQiso,ac);
       if not k in Aperm then
+        Add(Apa,k);
 	Aperm:=ClosureGroup(Aperm,k);
 	Add(A,InnerAutomorphism(Q,GeneratorsOfGroup(Q)[j]));
       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->InnerAutomorphism(Q,x));
     gens:=ShallowCopy(innB);
+    Info(InfoMorph,2,"|gens|=",Length(gens),"+",Length(C),
+      "+",Length(B),"+",Length(A));
     Append(gens,C);
     Append(gens,B);
     Append(gens,A);

diff --git a/lib/morpheus.gi b/lib/morpheus.gi
@@ -1,4 +1,5 @@
-#############################################################################
+#
+###########################################################################
 ##
 ##  This file is part of GAP, a system for computational discrete algebra.
 ##  This file's authors include Alexander Hulpke.
@@ -572,11 +573,12 @@ local hom, allinner, gens, c, ran, r, cen, img, dom, u, subs, orbs, cnt,
       o:=cl;
       cl:=[];
       for i in o do
-        if not ForAny(cl,x->Order(Representative(i))=Order(Representative(x)) 
-          and (Size(x) mod Size(i)=0) and Representative(i) in x) then
+# test to avoid duplicates is more expensive than it is worth
+#        if not ForAny(cl,x->Order(Representative(i))=Order(Representative(x)) 
+#          and (Size(x) mod Size(i)=0) and Representative(i) in x) then
           r:=ConjugacyClass(g,Representative(i));
           Add(cl,r);
-        fi;
+#        fi;
       od;
 
       Info(InfoMorph,2,"actbase ",Length(cl), " classes");

diff --git a/tst/teststandard/opers/AutomorphismGroup.tst b/tst/teststandard/opers/AutomorphismGroup.tst
@@ -26,17 +26,12 @@ true
 
 #
 gap> G:=GL(IsPermGroup,3,3);;
-gap> H:=AutomGrpSR(G);;
-gap> StructureDescription(H);
-"PSL(3,3) : C2"
-
-#
-gap> G:=GL(IsPermGroup,3,3);;
-gap> H:=AutomorphismGroupMorpheus(G);;
+gap> H:=AutomorphismGroup(G);;
 gap> StructureDescription(H);
 "PSL(3,3) : C2"
 
 #
+gap> SetAssertionLevel(0);
 gap> g:=PerfectGroup(IsPermGroup,15360,1);;
 gap> h:=g^(1,129);;
 gap> AutomorphismGroup(g);; # pull out of isom. test (reduce timeout risk)