Improve FastNormalClosure

Extend the first argument to accept either a group <G> or generators
<grpgens> of a group <G>.

Document the assumption, that the group generated by <grpgens>
is an overgroup of the one generated by <list>.

Add a test whether the group <G> is abelian.

Author: ssiccha

gap/generic/kernel.gi

@@ -150,20 +150,28 @@ InstallGlobalFunction( RandomSubproduct, function(a)
 end );
 
 # Computes randomly (it might underestimate) the normal closure of <list>
-# under conjugation by the group generated by <grpgens>. The integer
-# <n> controls how many new conjugates are computed.
+# under conjugation by <G>. <G> can be a group containing <list> or a
+# list of generators of such a group. The integer <n> controls how many new
+# conjugates are computed.
 # The error probability can be determined by following Theorem 2.3.9 in
-# <Cite Key="Ser03"/>.  According to Lemma 2.3.8, if <grpgens> has 4 or more
-# entries, then as long as the result does not generate the full normal closure
-# of <list> under <grpgens>, the probability that a conjugate is not contained
+# <Cite Key="Ser03"/>.  According to Lemma 2.3.8, if <G> has 4 or more
+# generators, then as long as the result does not generate the full normal closure
+# of <list> under <G>, the probability that a conjugate is not contained
 # in the group generated by the result is >= 1/4.
-InstallGlobalFunction( FastNormalClosure , function( grpgens, list, n )
-  local list2, fewGenerators, repetitions, randlist, conjugators, i, c;
+InstallGlobalFunction( FastNormalClosure , function( G, list, n )
+  local list2, grpgens, fewGenerators, repetitions, randlist, conjugators, i, c;
   if IsEmpty(list) then
     return [];
   fi;
   list2 := ShallowCopy(list);
-  if Length(grpgens) = 1 then
+  if IsGroup(G) then
+    grpgens := GeneratorsOfGroup(G);
+  else
+    grpgens := G;
+  fi;
+  if (IsGroup(G) and HasIsAbelian(G) and IsAbelian(G))
+      or (IsGroup(G) and HasIsCyclic(G) and IsCyclic(G))
+      or Length(grpgens) = 1 then
     return list2;
   fi;
   fewGenerators := Length(grpgens) <= 3;

gap/projective/almostsimple.gi

@@ -683,7 +683,7 @@ RECOG.simplesocle := function(ri,g)
     comm3:=Comm(comm2,comm2^y);
   until not ri!.isone(comm3);
 
-  gensH:=FastNormalClosure(GeneratorsOfGroup(g),[comm3],20);
+  gensH:=FastNormalClosure(g,[comm3],20);
 
   return gensH;
 end;

gap/projective/classicalnatural.gi

@@ -1469,7 +1469,7 @@ RECOG.GuessProjSL2ElmOrder := function(x,f)
   return o;
 end;
 
-RECOG.IsThisSL2Natural := function(gens,f)
+RECOG.IsThisSL2Natural := function(ri,gens,f)
   # Checks quickly whether or not this is SL(2,f).
   # The answer is not guaranteed to be correct, this is Las Vegas.
   local CheckElm,a,b,clos,coms,i,isabelian,j,l,notA5,p,q,S,seenqm1,seenqp1,x;
@@ -3193,7 +3193,7 @@ function(ri, g)
 
   # First check whether we are applicable:
   if d = 2 then
-      if not RECOG.IsThisSL2Natural(GeneratorsOfGroup(g),f) then
+      if not RECOG.IsThisSL2Natural(ri,GeneratorsOfGroup(g),f) then
           Info(InfoRecog,2,"ClassicalNatural: Is not PSL_2.");
           return fail; # FIXME: fail = TemporaryFailure here really correct?
       fi;

gap/projective/d247.gi

@@ -362,7 +362,7 @@ function(ri, G)
     # This is called with an element that we hope lies in a normal subgroup.
     local H,a,basis,collf,conjgensG,count,dim,hom,homcomp,homs,homsimg,i,
           kro,m,mm,mult,ngens,nngens,o,orb,pr,r,subdim,y,z;
-    ngens := FastNormalClosure(GeneratorsOfGroup(G),[x],4);
+    ngens := FastNormalClosure(G,[x],4);
     m := GModuleByMats(ngens,f);
     if MTX.IsIrreducible(m) then
         if not ispower then
@@ -440,7 +440,7 @@ function(ri, G)
   CheckNormalClosure := function(x)
     # This is called with an element that we hope lies in a normal subgroup.
     local m,ngens;
-    ngens := FastNormalClosure(GeneratorsOfGroup(G),x,4);
+    ngens := FastNormalClosure(G,x,4);
     m := GModuleByMats(ngens,f);
     if not IsIrreducible(m) then
         Info(InfoRecog,2,"Proto: Seem to have found something!");

gap/projective/findnormal.gi

@@ -743,7 +743,7 @@ function(ri,G)
       return fail; # FIXME: fail = TemporaryFailure here really correct?
   fi;
   if not IsBound(r.Nready) or not r.Nready then
-      r.Ngens := FastNormalClosure(GeneratorsOfGroup(G),[r.el],20);
+      r.Ngens := FastNormalClosure(G,[r.el],20);
   fi;
   f := ri!.field;
   m := GModuleByMats(r.Ngens,f);

gap/projective/tensor.gi

@@ -17,7 +17,7 @@
 ##
 #############################################################################
 
-RECOG.FindTensorKernel := function(G,onlyone)
+RECOG.FindTensorKernel := function(ri,G,onlyone)
   # Assume G respects a tensor product decomposition of its natural
   # module V. Try to find the kernel of the canonical map:
   local N,allps,c,fac,facs,i,j,kgens,newc,notused,o,pfacs,x,z;
@@ -60,7 +60,7 @@ RECOG.FindTensorKernel := function(G,onlyone)
           fi;
       od;
       #Print(Length(notused)," \c");
-      N := GroupWithGenerators(FastNormalClosure(GeneratorsOfGroup(G),[c],10));
+      N := GroupWithGenerators(FastNormalClosure(G,[c],10));
       if onlyone and
          (ForAny(GeneratorsOfGroup(N),m->IsZero(m[1,1]) or
                                          not IsOne(m*(m[1,1])^-1))) then
@@ -339,7 +339,7 @@ function(ri,G)
 
   # Now assume a tensor factorization exists:
   #Gm := GroupWithMemory(G);???
-  N := RECOG.FindTensorKernel(G,true);
+  N := RECOG.FindTensorKernel(ri,G,true);
   Info(InfoRecog,3,
        "TensorDecomposable: I seem to have found a normal subgroup...");
   r := RECOG.FindTensorDecomposition(G,N);

misc/colva/AlmostSimple.g

@@ -64,7 +64,7 @@ StableDerivative := function(G)
        r := PseudoRandomNormalClosureElement(G,H);
        Add(gens,Ggens[i]^-1*r^-1*Ggens[i]*r);
    od;
-   gens := FastNormalClosure(Ggens,gens,1);
+   gens := FastNormalClosure(G,gens,1);
    H := GroupWithGenerators(gens);
    G := ShallowCopy(H);
   until IsProbablyPerfect(G);

misc/colva/DPleaf.g

@@ -308,7 +308,7 @@ SolveLeafDP := function(ri,rifac,name)
  econj := List([1..Size(e)],i->GroupHomomorphismByFunction(Grp(ri),Grp(ri),g->g^e[i]));
 
  gens := List([1..3],i->FindPoint(Grp(ri),phi,1,I));
- H1 := SubgroupNC(Grp(ri),FastNormalClosure(GeneratorsOfGroup(Grp(ri)),gens,1));
+ H1 := SubgroupNC(Grp(ri),FastNormalClosure(Grp(ri),gens,1));
  H1toblk := phi*projs[1];
  H1toblk!.Source := H1;
  blk := GroupWithGenerators(List(GeneratorsOfGroup(H1),x->ImageElm(H1toblk,x)));

misc/colva/NSM.g

@@ -357,7 +357,7 @@ LowIndexMaps := function(G,ri)
  kgens := List([1..15],i->PseudoRandom(G));
  kims := List(kgens,x->ImageElm(rho,x));
  kgens := List([1..15],i->kgens[i]*ImageElm(rhoinv,kims[i])^-1);
- kgens := FastNormalClosure(GeneratorsOfGroup(G),kgens,3);
+ kgens := FastNormalClosure(Grp(ri),kgens,3);
 
 # Compute the stabilizer of 1 in P
  s1gens := List(GeneratorsOfGroup(Stabiliser(P,1)),x->ImageElm(rhoinv,x));

misc/colva/SimpleRecog.g

@@ -513,7 +513,7 @@ RecogniseQuasiSimpleDP := function(G,H,phi,I,name)
 
 # Find generators for one block
  gens := List([1..3],i->FindPoint(H,phi,1,I));
- H1 := SubgroupNC(H,FastNormalClosure(GeneratorsOfGroup(H),gens,1));
+ H1 := SubgroupNC(H,FastNormalClosure(Grp(ri),gens,1));
  H1toblk := phi*projs[1];
  H1toblk!.Source := H1;
  blk := GroupWithGenerators(List(GeneratorsOfGroup(H1),x->ImageElm(H1toblk,x)));