Add DirectProductOfPermGroupsWithMovedPoints helper

This is basically what DirectProduct does for a set of permutation groups,
except that one can override the moved points for each factor. This is useful
if one needs to compute "sums of G-sets": If e.g. we have a trivial group
acting on 3 points, and a Sym(4), then DirectProduct just gives back the
Sym(4). But if we need the result to act on 3+4=7 points, with the Sym(4)
acting on points acting on [4..7], then this new helper allows us to do that.

lib/gprdperm.gi

@@ -12,21 +12,13 @@
 
 #############################################################################
 ##
-#M  DirectProductOp( <grps>, <G> )  . . . . . . direct product of perm groups
+#F  DirectProductOfPermGroupsWithMovedPoints( <grps>, <pnts> )
 ##
-InstallMethod( DirectProductOp,
-    "for a list of permutation groups, and a permutation group",
-    IsCollsElms,
-    [ IsList and IsPermCollColl, IsPermGroup ], 0,
-    function( grps, G )
+BindGlobal("DirectProductOfPermGroupsWithMovedPoints",
+    function( grps, pnts )
     local   oldgrps,  olds,  news,  perms,  gens,
             deg,  grp,  old,  new,  perm,  gen,  D, info;
 
-    # Check the arguments.
-    if not ForAll( grps, IsGroup ) then
-      TryNextMethod();
-    fi;
-
     oldgrps := [  ];
     olds    := [  ];
     news    := [  ];
@@ -35,10 +27,11 @@ InstallMethod( DirectProductOp,
     deg     := 0;
 
     # loop over the groups
-    for grp  in grps  do
+    for i in Length(grps)  do
 
         # find old domain, new domain, and conjugating permutation
-        old  := MovedPoints( grp );
+        grp  := grps[i];
+        old  := pnts[i]l
         new  := [deg+1..deg+Length(old)];
         perm := MappingPermListList( old, new );
         deg  := deg + Length(old);
@@ -66,6 +59,25 @@ InstallMethod( DirectProductOp,
     end );
 
 
+#############################################################################
+##
+#M  DirectProductOp( <grps>, <G> )  . . . . . . direct product of perm groups
+##
+InstallMethod( DirectProductOp,
+    "for a list of permutation groups, and a permutation group",
+    IsCollsElms,
+    [ IsList and IsPermCollColl, IsPermGroup ], 0,
+    function( grps, G )
+
+    # Check the arguments.
+    if not ForAll( grps, IsGroup ) then
+      TryNextMethod();
+    fi;
+
+    return DirectProductOfPermGroupsWithMovedPoints(grps, List(grps, MovedPoints));
+    end );
+
+
 #############################################################################
 ##
 #M  Size( <D> ) . . . . . . . . . . . . . . . . . . . . . . of direct product

tst/testinstall/gprd.tst

@@ -45,5 +45,12 @@ gap> d4 := DirectProduct(g1, g2, g3);;
 gap> HasIsFinite(d4) and not IsFinite(d4);
 true
 
+#
+gap> d1 := DirectProductOfPermGroupsWithMovedPoints([g1, g1, g1], [[1..4], [1..4], [1..4]]);;
+gap> MovedPoints(d1);
+[ 1, 2, 3, 5, 6, 7, 9, 10, 11 ]
+gap> HasIsFinite(g1) and IsFinite(g1);
+true
+
 #
 gap> STOP_TEST("gprd.tst");