Add IsAbelian filters to IsSolvableGroup and DerivedSeries

- IsAbelian filters are added to IsSolvableGroup and DerivedSeries.
- IsSolvable filters are added to DerivedSeriesOfGroup.
- @hulpke's enhancement on DerivedSeriesOfGroup in commit e5eb173
  about HasGeneratorsOfGroup and about HasAbelianInvariants are enhanced
  with HasIsTrivial and HasIsPerfectGroup.

lib/grp.gi

@@ -412,6 +412,10 @@ InstallImmediateMethod( IsPerfectGroup,
  0,
  IsTrivial );
 
+InstallMethod( IsPerfectGroup, "for groups having abelian invariants",
+ [ IsGroup and HasAbelianInvariants ],
+ grp -> Length( AbelianInvariants( grp ) ) = 0 );
+
 InstallMethod( IsPerfectGroup,
  "method for finite groups",
  [ IsGroup and IsFinite ],
@@ -528,13 +532,22 @@ InstallMethod( IsSolvableGroup,
  "generic method for groups",
  [ IsGroup ],
  function ( G )
- local S; # derived series of <G>
+ local S, # derived series of <G>
+ isAbelian, # true if <G> is abelian
+ isSolvable; # true if <G> is solvable
 
  # compute the derived series of <G>
  S := DerivedSeriesOfGroup( G );
 
  # the group is solvable if the derived series reaches the trivial group
- return IsTrivial( S[ Length( S ) ] );
+ isSolvable := IsTrivial( S[ Length( S ) ] );
+
+ # set IsAbelian filter
+ isAbelian := isSolvable and Length( S ) <= 2;
+ Assert(3, IsAbelian(G) = isAbelian);
+ SetIsAbelian(G, isAbelian);
+
+ return isSolvable;
  end );
 
 
@@ -895,16 +908,35 @@ InstallMethod( DerivedSeriesOfGroup,
  D := DerivedSubgroup( G );
 
  while
- # we don't know that the group has no generators
- (not HasGeneratorsOfGroup(S[Length(S)]) or
- Length(GeneratorsOfGroup(S[Length(S)]))>0) and
- ( (not HasAbelianInvariants(S[Length(S)]) and D <> S[ Length(S) ]) or
- Length(AbelianInvariants(S[Length(S)]))>0) do
+ (not HasIsTrivial(S[Length(S)]) or
+ not IsTrivial(S[Length(S)])) and
+ (
+ (not HasIsPerfectGroup(S[Length(S)]) and
+ not HasAbelianInvariants(S[Length(S)]) and D <> S[ Length(S) ]) or
+ (HasIsPerfectGroup(S[Length(S)]) and not IsPerfectGroup(S[Length(S)]))
+ or (HasAbelianInvariants(S[Length(S)])
+ and Length(AbelianInvariants(S[Length(S)])) > 0)
+ ) do
  Add( S, D );
  Info( InfoGroup, 2, "DerivedSeriesOfGroup: step ", Length(S) );
  D := DerivedSubgroup( D );
  od;
 
+ # set filters if the last term is known to be trivial
+ if HasIsTrivial(S[Length(S)]) and IsTrivial(S[Length(S)]) then
+ SetIsSolvableGroup(G, true);
+ if Length(S) <=2 then
+ Assert(2, IsAbelian(G));
+ SetIsAbelian(G, true);
+ fi;
+ fi;
+
+ # set IsAbelian filter if length of derived series is more than 2
+ if Length(S) > 2 then
+ Assert(2, not IsAbelian(G));
+ SetIsAbelian(G, false);
+ fi;
+
  # return the series when it becomes stable
  return S;
  end );

tst/testinstall/opers/IsSolvableGroup.tst

@@ -0,0 +1,56 @@
+gap> START_TEST("IsSolvableGroup.tst");
+gap> List(AllGroups(120), IsSolvableGroup);
+[ true, true, true, true, false, true, true, true, true, true, true, true, 
+ true, true, true, true, true, true, true, true, true, true, true, true, 
+ true, true, true, true, true, true, true, true, true, false, false, true, 
+ true, true, true, true, true, true, true, true, true, true, true ]
+gap> List(AllTransitiveGroups(DegreeAction, 8), IsSolvable);
+[ true, true, true, true, true, true, true, true, true, true, true, true, 
+ true, true, true, true, true, true, true, true, true, true, true, true, 
+ true, true, true, true, true, true, true, true, true, true, true, true, 
+ false, true, true, true, true, true, false, true, true, true, true, false, 
+ false, false ]
+gap> IsSolvable(DihedralGroup(24));
+true
+gap> IsSolvable(DihedralGroup(IsFpGroup,24));
+true
+gap> DerivedSeries(Group(()));
+[ Group(()) ]
+gap> G := Group((6,7,8,9,10),(8,9,10),(1,2)(6,7),(1,2,3,4)(6,7,8,9));;
+gap> Length(DerivedSeriesOfGroup(G));
+4
+gap> HasIsSolvableGroup(G) and not IsSolvable(G) and HasIsAbelian(G) and not IsAbelian(G);
+true
+gap> IsSolvableGroup(AbelianGroup([2,3,4,5,6,7,8,9,10]));
+true
+gap> HasIsSolvableGroup(AbelianGroup(IsFpGroup,[2,3,4,5,6,7,8,9,10]));
+true
+gap> IsSolvableGroup(AbelianGroup(IsFpGroup,[2,3,4,5,6,7,8,9,10]));
+true
+gap> IsSolvableGroup(Group(()));
+true
+gap> A := AbelianGroup([3,3,3]);; H := AutomorphismGroup(A);;
+gap> B := SylowSubgroup(H, 13);; G := SemidirectProduct(B, A);;
+gap> HasIsSolvableGroup(G) and IsSolvable(G);
+true
+
+## some fp-groups
+## The following four tests check whether the current IsSolvable method using
+## DerivedSeriesOfGroup indeed adds IsAbelian whenever it is appropriate. If
+## later new methods for IsSolvable are added, these tests may fail. Then
+## these four tests need to be modified accordingly.
+gap> F := FreeGroup("r", "s");; r := F.1;; s := F.2;;
+gap> G := F/[s^2, s*r*s*r];;
+gap> IsSolvable(G) and HasIsAbelian(G) and not IsAbelian(G);
+true
+gap> F := FreeGroup("a", "b", "c", "d");; a := F.1;; b := F.2;; c := F.3;; d:= F.4;;
+gap> G := F/[ a^2, b^2, a*b*a^(-1)*b^(-1), c, d ];;
+gap> IsSolvable(G) and HasIsAbelian(G) and IsAbelian(G);
+true
+gap> G := F/[ a^2, b^2, c^2, (a*b)^3, (b*c)^3, (a*c)^2, d ];;
+gap> IsSolvable(G) and HasIsAbelian(G) and not IsAbelian(G);
+true
+gap> G := F/[ a^2, b^2, c^2, d^2, (a*b)^3, (b*c)^3, (c*d)^3, (a*c)^2, (a*d)^2, (b*d)^2 ];;
+gap> not IsSolvable(G) and HasIsAbelian(G) and not IsAbelian(G);
+true
+gap> STOP_TEST("IsSolvableGroup.tst", 10000);