Merge pull request #400 from fingolfin/mh/fit-frat-nilpotent

Mark FittingSubgroup and FrattiniSubgroup as nilpotent

lib/grp.gi

@@ -990,19 +990,20 @@ InstallMethod( Exponent,
 #M  FittingSubgroup( <G> )  . . . . . . . . . . . Fitting subgroup of a group
 ##
 InstallMethod( FittingSubgroup, "for nilpotent group",
-    [ IsGroup and IsNilpotentGroup ], 0, IdFunc );
+    [ IsGroup and IsNilpotentGroup ], SUM_FLAGS, IdFunc );
 
 InstallMethod( FittingSubgroup,
     "generic method for groups",
     [ IsGroup and IsFinite ],
     function (G)
-        if IsTrivial (G) then
-            return G;
-        else
-            return SubgroupNC( G, Filtered(Union( List( Set( FactorsInt( Size( G ) ) ),
+        if not IsTrivial( G ) then
+            G := SubgroupNC( G, Filtered(Union( List( Set( FactorsInt( Size( G ) ) ),
                          p -> GeneratorsOfGroup( PCore( G, p ) ) ) ),
                          p->p<>One(G)));
+            Assert( 2, IsNilpotentGroup( G ) );
+            SetIsNilpotentGroup( G, true );
         fi;
+        return G;
     end);
 
 RedispatchOnCondition( FittingSubgroup, true, [IsGroup], [IsFinite], 0);
@@ -1019,7 +1020,12 @@ local m;
       return G;
     fi;
     m:=List(ConjugacyClassesMaximalSubgroups(G),C->Core(G,Representative(C)));
-    return Intersection(m);
+    m := Intersection(m);
+    if HasIsFinite(G) and IsFinite(G) then
+      Assert(2,IsNilpotentGroup(m));
+      SetIsNilpotentGroup(m,true);
+    fi;
+    return m;
 end);
 
 

lib/grppcatr.gi

@@ -134,7 +134,10 @@ InstallMethod( FrattiniSubgroup,
     0,
 
 function( G )
-    return Core( G, PrefrattiniSubgroup( G ) );
+    G := Core( G, PrefrattiniSubgroup( G ) );
+    Assert( 2, IsNilpotentGroup( G) );
+    SetIsNilpotentGroup( G, true );
+    return G;
 end);
 
 

lib/maxsub.gi

@@ -346,6 +346,9 @@ local m,f,i;
       f:=Core(G,NormalIntersection(f,m[i]));
     fi;
   od;
+  if HasIsFinite(G) and IsFinite(G) then
+    SetIsNilpotentGroup(f,true);
+  fi;
   return f;
 end);
 

tst/testinstall/opers/FittingSubgroup.tst

@@ -0,0 +1,48 @@
+gap> START_TEST("FittingSubgroup.tst");
+
+#
+gap> G:=SylowSubgroup(SymmetricGroup(5),2);;
+gap> HasIsNilpotentGroup(G);
+true
+gap> IsIdenticalObj(G, FittingSubgroup(G));
+true
+
+#
+gap> G := CyclicGroup(IsPermGroup, 12);;
+gap> IsIdenticalObj(G, FittingSubgroup(G));
+true
+gap> G := CyclicGroup(IsPcGroup, 12);;
+gap> IsIdenticalObj(G, FittingSubgroup(G));
+true
+
+#
+gap> List(AllSmallGroups(60), g -> Size(FittingSubgroup(g)));
+[ 30, 30, 30, 60, 1, 15, 15, 15, 20, 30, 30, 30, 60 ]
+gap> ForAll(AllSmallGroups(60), g -> IsNormal(g, FrattiniSubgroup(g)));
+true
+
+#
+gap> g := SL(2,5);;
+gap> f := FittingSubgroup(g);; Size(f);
+2
+gap> HasIsNilpotentGroup(f);
+true
+gap> p := SylowSubgroup(g, 2);;
+gap> HasIsNilpotentGroup(p);
+true
+gap> HasIsNilpotentGroup(FittingSubgroup(p));
+true
+
+#
+gap> g := SL(IsPermGroup,2,5);;
+gap> f := FittingSubgroup(g);;
+gap> HasIsNilpotentGroup(f);
+true
+gap> p := SylowSubgroup(g, 2);;
+gap> HasIsNilpotentGroup(p);
+true
+gap> HasIsNilpotentGroup(FittingSubgroup(p));
+true
+
+#
+gap> STOP_TEST("FittingSubgroup.tst", 1);

tst/testinstall/opers/FrattiniSubgroup.tst

@@ -0,0 +1,51 @@
+gap> START_TEST("FrattiniSubgroup.tst");
+
+#
+gap> FrattiniSubgroup(SymmetricGroup(3));
+Group(())
+gap> FrattiniSubgroup(SymmetricGroup(4));
+Group(())
+gap> FrattiniSubgroup(SymmetricGroup(5));
+Group(())
+
+#
+gap> FrattiniSubgroup(CyclicGroup(IsPermGroup, 3));
+Group(())
+gap> FrattiniSubgroup(CyclicGroup(IsPermGroup, 9));
+Group([ (1,4,7)(2,5,8)(3,6,9) ])
+gap> FrattiniSubgroup(CyclicGroup(IsPcGroup, 3));
+Group([  ])
+gap> FrattiniSubgroup(CyclicGroup(IsPcGroup, 9));
+Group([ f2 ])
+
+#
+gap> List(AllSmallGroups(60), g -> Size(FrattiniSubgroup(g)));
+[ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
+gap> ForAll(AllSmallGroups(60), g -> IsNormal(g, FrattiniSubgroup(g)));
+true
+
+#
+gap> g := SL(2,5);;
+gap> f := FrattiniSubgroup(g);
+<group of 2x2 matrices of size 2 over GF(5)>
+gap> HasIsNilpotentGroup(f);
+true
+gap> p := SylowSubgroup(g, 2);;
+gap> HasIsNilpotentGroup(p);
+true
+gap> HasIsNilpotentGroup(FrattiniSubgroup(p));
+true
+
+#
+gap> g := SL(IsPermGroup,2,5);;
+gap> f := FrattiniSubgroup(g);;
+gap> HasIsNilpotentGroup(f);
+true
+gap> p := SylowSubgroup(g, 2);;
+gap> HasIsNilpotentGroup(p);
+true
+gap> HasIsNilpotentGroup(FrattiniSubgroup(p));
+true
+
+#
+gap> STOP_TEST("FrattiniSubgroup.tst", 1);