Add new method for IsSolvableGroup

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> Set(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

lib/ctbl.gd

@@ -4103,10 +4103,10 @@ DeclareOperation( "CharacterTableIsoclinic",
 ##  gap> tg:= CharacterTable( g );;
 ##  gap> IsRecord(
 ##  >        TransformingPermutationsCharacterTables( index2[1], tg ) );
-##  false
+##  true
 ##  gap> IsRecord(
 ##  >        TransformingPermutationsCharacterTables( index2[2], tg ) );
-##  true
+##  false
 ##  ]]></Example>
 ##  <P/>
 ##  Alternatively, we could construct the character table of the central

lib/grp.gi

@@ -563,6 +563,10 @@ InstallMethod( IsAlmostSimpleGroup,
 ##  By the odd order theorem, N is solvable, and so is G. Thus the order of
 ##  any non-solvable finite group is a multiple of 4.
 ##
+##  By Burnside's theorem, every group of order p^a q^b is solvable. If a
+##  group of such order is not already caught by the reasoning above, then
+##  it must have order 2^a q^b with a>1.
+##
 InstallImmediateMethod( IsSolvableGroup, IsGroup and HasSize, 10,
     function( G )
     local size;
@@ -573,6 +577,32 @@ InstallImmediateMethod( IsSolvableGroup, IsGroup and HasSize, 10,
     TryNextMethod();
     end );
 
+InstallMethod( IsSolvableGroup,
+    "if group size is known and is not divisible by 4 or p^a q^b",
+    [ IsGroup and HasSize ], 25,
+    function( G )
+    local size;
+    size := Size( G );
+    if IsInt( size ) then
+      if size mod 4 <> 0 then
+        return true;
+      else
+        size := size/4;
+        while size mod 2 = 0 do
+          size := size/2;
+        od;
+        if size = 1 then
+          SetIsPGroup( G, true );
+          SetPrimePGroup( G, 2 );
+          return true;
+        elif IsPrimePowerInt( size ) then
+          return true;
+        fi;
+      fi;
+    fi;
+    TryNextMethod();
+    end );
+
 InstallMethod( IsSolvableGroup,
     "generic method for groups",
     [ IsGroup ],

lib/morpheus.gd

@@ -509,7 +509,7 @@ DeclareGlobalFunction("IsomorphismGroups");
 ##  gap> h:=Group((1,2,3),(1,2));
 ##  Group([ (1,2,3), (1,2) ])
 ##  gap> quo:=GQuotients(g,h);
-##  [ [ (1,3,2,4), (1,2,3) ] -> [ (2,3), (1,2,3) ] ]
+##  [ [ (1,2,3,4), (1,3,4) ] -> [ (2,3), (1,2,3) ] ]
 ##  ]]></Example>
 ##  </Description>
 ##  </ManSection>

lib/teaching.g

@@ -197,9 +197,9 @@ DeclareGlobalFunction("CosetDecomposition");
 ##  <A>H</A>-conjugacy.
 ##  <Example><![CDATA[
 ##  gap> AllHomomorphismClasses(SymmetricGroup(4),SymmetricGroup(3)); 
-##  [ [ (1,2,3,4), (1,2) ] -> [ (), () ],
-##    [ (1,2,3,4), (1,2) ] -> [ (2,3), (1,2) ],
-##    [ (1,2,3,4), (1,2) ] -> [ (1,2), (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>

tst/testinstall/opers/IsSolvableGroup.tst

@@ -33,6 +33,12 @@ 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
+gap> G := DirectProduct(CyclicGroup(27), SymmetricGroup(3));;
+gap> IsSolvableGroup(G);
+true
+gap> G := DirectProduct(CyclicGroup(6), SymmetricGroup(4));;
+gap> IsSolvableGroup(G);
+true
 
 ## some fp-groups
 ## The following four tests check whether the current IsSolvable method using
@@ -53,4 +59,13 @@ 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> 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 ];; Size(G);;
+gap> not IsSolvable(G) and not IsAbelian(G);
+true
+gap> F := FreeGroup("a", "x");; a := F.1;; x := F.2;;
+gap> G := F/[x^2*a^8, a^16, x*a*x^(-1)*a];; Size(G);;
+gap> IsSolvableGroup(G) and IsPGroup(G) and IsNilpotentGroup(G);
+true
+gap> PrimePGroup(G);
+2
 gap> STOP_TEST("IsSolvableGroup.tst", 1);