ENHANCE: Performance improvements to automorphism group computations.

Includes improvements to `SmallGeneratingSet` for perm groups.

Also dealt with changed generators in manual examples

Author: hulpke

doc/tut/group.xml

@@ -833,9 +833,9 @@ usual way we now look for the subgroups above <C>u105</C>.
 gap> blocks := Blocks( a8, orb );; Length( blocks );
 15
 gap> blocks[1];
-[ (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,8)(6,7), (1,4)(2,3)(5,7)(6,8), 
-  (1,5)(2,6)(3,8)(4,7), (1,6)(2,5)(3,7)(4,8), (1,7)(2,8)(3,6)(4,5), 
-  (1,8)(2,7)(3,5)(4,6) ]
+[ (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,4)(2,3)(5,8)(6,7),
+  (1,5)(2,6)(3,7)(4,8), (1,6)(2,5)(3,8)(4,7), (1,7)(2,8)(3,5)(4,6),
+  (1,8)(2,7)(3,6)(4,5) ]
 ]]></Example>
 <P/>
 To find the subgroup of index 15 we  again use closure. Now  we must be a
@@ -1175,8 +1175,8 @@ gap> aut := AutomorphismGroup( p );; NiceMonomorphism(aut);;
 gap> niceaut := NiceObject( aut );
 Group([ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ])
 gap> IsomorphismGroups( niceaut, SymmetricGroup( 4 ) );
-[ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ] -> 
-[ (1,4,3,2), (1,4,2), (1,3)(2,4), (1,4)(2,3) ]
+[ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ] ->
+[ (1,4,2,3), (1,2,3), (1,2)(3,4), (1,3)(2,4) ]
 ]]></Example>
 <P/>
 The range of  a nice monomorphism is  in most cases a permutation  group,

lib/autsr.gi

@@ -941,7 +941,11 @@ local d,a,map,possibly,cG,cH,nG,nH,i,j,sel,u,v,asAutomorphism,K,L,conj,e1,e2,
     u:=ClosureGroup(i,K);
     v:=ClosureGroup(i,L);
     if u<>v then
-      gens:=SmallGeneratingSet(api);
+      if IsSolvableGroup(api) then
+        gens:=Pcgs(api);
+      else
+        gens:=SmallGeneratingSet(api);
+      fi;
       pre:=List(gens,x->PreImagesRepresentative(iso,x));
       map:=RepresentativeAction(SubgroupNC(a,pre),u,v,asAutomorphism);
       if map=fail then

lib/ctbl.gd

@@ -4380,11 +4380,11 @@ DeclareGlobalFunction( "NormalSubgroupClasses" );
 ##    Character( CharacterTable( S4 ), [ 3, 1, -1, 0, -1 ] ), 
 ##    Character( CharacterTable( S4 ), [ 1, 1, 1, 1, 1 ] ) ]
 ##  gap> kernel:= KernelOfCharacter( irr[3] );
-##  Group([ (1,2)(3,4), (1,3)(2,4) ])
+##  Group([ (1,2)(3,4), (1,4)(2,3) ])
 ##  gap> HasNormalSubgroupClassesInfo( tbl );
 ##  true
 ##  gap> NormalSubgroupClassesInfo( tbl );
-##  rec( nsg := [ Group([ (1,2)(3,4), (1,3)(2,4) ]) ], 
+##  rec( nsg := [ Group([ (1,2)(3,4), (1,4)(2,3) ]) ],
 ##    nsgclasses := [ [ 1, 3 ] ], nsgfactors := [  ] )
 ##  gap> ClassPositionsOfNormalSubgroup( tbl, kernel );
 ##  [ 1, 3 ]

lib/grpperm.gi

@@ -1892,10 +1892,20 @@ end);
 InstallMethod(SmallGeneratingSet,"random and generators subset, randsims",true,
   [IsPermGroup],0,
 function (G)
-local  i, j, U, gens,o,v,a,sel,min;
+local  i, j, U, gens,o,v,a,sel,min,orb,orp,ok;
 
-  # remove obvious redundancies
   gens := ShallowCopy(Set(GeneratorsOfGroup(G)));
+
+  # try pc methods first. The solvability test should not exceed cost, nor
+  # the number of points.
+  if #Length(MovedPoints(G))<50000 and 
+   #((HasIsSolvableGroup(G) and IsSolvableGroup(G)) or IsAbelian(G))
+      IsSolvableGroup(G) 
+      and Length(gens)>3 then
+    return MinimalGeneratingSet(G);
+  fi;
+
+  # remove obvious redundancies
   o:=List(gens,Order);
   SortParallel(o,gens,function(a,b) return a>b;end);
   sel:=Filtered([1..Length(gens)],x->o[x]>1);
@@ -1920,26 +1930,32 @@ local  i, j, U, gens,o,v,a,sel,min;
   od;
   gens:=gens{sel};
 
-  # try pc methods first
-  if Length(MovedPoints(G))<1000 and HasIsSolvableGroup(G) 
-     and IsSolvableGroup(G) and Length(gens)>3 then
-    return MinimalGeneratingSet(G);
-  fi;
-
+  # store orbit data
+  orb:=Set(List(Orbits(G,MovedPoints(G)),Set));
+  orp:=Filtered([1..Length(orb)],x->IsPrimitive(Action(G,orb[x])));
 
   min:=2;
   if Length(gens)>2 then
   # minimal: AbelianInvariants
     min:=Maximum(List(Collected(Factors(Size(G)/Size(DerivedSubgroup(G)))),x->x[2]));
+    if min=Length(GeneratorsOfGroup(G)) then return GeneratorsOfGroup(G);fi;
     i:=Maximum(2,min);
     while i<=min+1 and i<Length(gens) do
       # try to find a small generating system by random search
       j:=1;
       while j<=5 and i<Length(gens) do
         U:=Subgroup(G,List([1..i],j->Random(G)));
+        ok:=true;
+        # first test orbits
+        if ok then
+          ok:=Length(orb)=Length(Orbits(U,MovedPoints(U))) and
+              ForAll(orp,x->IsPrimitive(U,orb[x]));
+        fi;
+
+
 	StabChainOptions(U).random:=100; # randomized size
 #Print("A:",i,",",j," ",Size(G)/Size(U),"\n");
-        if Size(U)=Size(G) then
+        if ok and Size(U)=Size(G) then
           gens:=Set(GeneratorsOfGroup(U));
         fi;
         j:=j+1;
@@ -1955,6 +1971,14 @@ local  i, j, U, gens,o,v,a,sel,min;
   while i <= Length(gens) and Length(gens)>min do
     # random did not improve much, try subsets
     U:=Subgroup(G,gens{Difference([1..Length(gens)],[i])});
+
+    ok:=true;
+    # first test orbits
+    if ok then
+      ok:=Length(orb)=Length(Orbits(U,MovedPoints(U))) and
+          ForAll(orp,x->IsPrimitive(U,orb[x]));
+    fi;
+
     StabChainOptions(U).random:=100; # randomized size
 #Print("B:",i," ",Size(G)/Size(U),"\n");
     if Size(U)<Size(G) then

lib/teaching.g

@@ -194,9 +194,9 @@ DeclareGlobalFunction("CosetDecomposition");
 ##  <A>H</A>-conjugacy.
 ##  <Example><![CDATA[
 ##  gap> AllHomomorphismClasses(SymmetricGroup(4),SymmetricGroup(3)); 
-##  [ [ (2,4,3), (1,2,3,4) ] -> [ (), () ], 
-##    [ (2,4,3), (1,2,3,4) ] -> [ (), (1,2) ], 
-##    [ (2,4,3), (1,2,3,4) ] -> [ (1,2,3), (1,2) ] ]
+##  [ [ (1,2,3), (2,4) ] -> [ (), () ],
+##    [ (1,2,3), (2,4) ] -> [ (), (1,2) ],
+##    [ (1,2,3), (2,4) ] -> [ (1,2,3), (1,2) ] ]
 ##  ]]></Example>
 ##  </Description>
 ##  </ManSection>