Enhance HasXXFactorGroup

HasSolvableFactorGroup computes commutators iteratively, and checks if
they are in the normal subgroup. If DerivedSeries is already known,
then checks if its last term is in N.
HasAbelianFactorGroup and HasElementaryAbelianFactorGroup returns true if
group is already known to be abelian or elementary abelian. The latter
previously errored out if normal subgroup was the whole group, fixed.
Tests are added.

lib/grp.gi

@@ -5059,6 +5059,9 @@ InstallMethod( KnowsHowToDecompose,
 ##
 InstallGlobalFunction(HasAbelianFactorGroup,function(G,N)
 local gen;
+  if HasIsAbelian(G) and IsAbelian(G) then
+    return true;
+  fi;
   Assert(2,IsNormal(G,N) and IsSubgroup(G,N));
   gen:=Filtered(GeneratorsOfGroup(G),i->not i in N);
   return ForAll([1..Length(gen)],
@@ -5070,11 +5073,29 @@ end);
 #M  HasSolvableFactorGroup(<G>,<N>)   test whether G/N is solvable
 ##
 InstallGlobalFunction(HasSolvableFactorGroup,function(G,N)
-local s,l;
+local gen, D, s, l;
+
+  if HasIsSolvableGroup(G) and IsSolvableGroup(G) then
+    return true;
+  fi;
   Assert(2,IsNormal(G,N) and IsSubgroup(G,N));
-  s := DerivedSeriesOfGroup(G);
-  l := Length(s);
-  return IsSubgroup(N,s[l]);
+  if HasDerivedSeriesOfGroup(G) then
+    s := DerivedSeriesOfGroup(G);
+    l := Length(s);
+    return IsSubgroup(N,s[l]);
+  fi;
+  D := G;
+  repeat
+    gen:=Filtered(GeneratorsOfGroup(D),i->not i in N);
+    if ForAll([1..Length(gen)],
+                i->ForAll([1..i-1],j->Comm(gen[i],gen[j]) in N)) then
+      return true;
+    fi;
+    D := DerivedSubgroup(D);
+  until IsPerfectGroup(D);
+  # this may be dangerous if N does not contain the identity of G
+  SetIsSolvableGroup(G, false);
+  return false;
 end);
 
 #############################################################################
@@ -5083,10 +5104,16 @@ end);
 ##
 InstallGlobalFunction(HasElementaryAbelianFactorGroup,function(G,N)
 local gen,p;
+  if HasIsElementaryAbelian(G) and IsElementaryAbelian(G) then
+    return true;
+  fi;
   if not HasAbelianFactorGroup(G,N) then
     return false;
   fi;
   gen:=Filtered(GeneratorsOfGroup(G),i->not i in N);
+  if gen = [] then
+    return true;
+  fi;
   p:=First([2..Order(gen[1])],i->gen[1]^i in N);
   return IsPrime(p) and ForAll(gen{[2..Length(gen)]},i->i^p in N);
 end);

tst/testinstall/grpperm.tst

@@ -0,0 +1,62 @@
+gap> START_TEST("grpperm.tst");
+gap> G := Group((1,2),(1,2,3,4));;
+gap> HasAbelianFactorGroup(G,G);
+true
+gap> HasElementaryAbelianFactorGroup(G,G);
+true
+gap> HasSolvableFactorGroup(G,G);
+true
+gap> N := Group((1,2)(3,4),(1,3)(2,4));;
+gap> HasAbelianFactorGroup(G,N);
+false
+gap> HasElementaryAbelianFactorGroup(G,N);
+false
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> IsAbelian(N);
+true
+gap> HasAbelianFactorGroup(N, Group((1,2)));
+true
+gap> IsElementaryAbelian(N);
+true
+gap> HasElementaryAbelianFactorGroup(N, Group((1,2)));
+true
+gap> N := Group((1,2)(3,4),(1,2,3));;
+gap> HasAbelianFactorGroup(G,N);
+true
+gap> HasElementaryAbelianFactorGroup(G,N);
+true
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> IsSolvable(G);
+true
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> F := FreeGroup("x","y");; x := F.1;; y:=F.2;;
+gap> G := F/[x^18,y];;
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^2));
+true
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^3));
+true
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^6));
+false
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^9));
+false
+gap> HasSolvableFactorGroup(SymmetricGroup(5), Group(()));
+false
+gap> HasSolvableFactorGroup(SymmetricGroup(5), SymmetricGroup(5));
+true
+gap> HasSolvableFactorGroup(SymmetricGroup(5), AlternatingGroup(5));
+true
+gap> HasSolvableFactorGroup(AlternatingGroup(5), AlternatingGroup(5));
+true
+gap> cube := Group(( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) );;
+gap> HasSolvableFactorGroup(cube, Center(cube));
+false
+gap> HasSolvableFactorGroup(cube, DerivedSeriesOfGroup(cube)[2]);
+true
+gap> G := SylowSubgroup(SymmetricGroup(2^7),2);;
+gap> N := Center(G);;
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> STOP_TEST( "grpperm.tst", 1);