Dear forum, Dear Gary McConnell, On Nov 30, 2011, at 11/30/11 11:27, Gary McConnell wrote:
> Hi Everyone > > Let G be a group with two subgroups H,K and consider the double coset space > DC = H\G/K. > > Denote by N_G(K) the normalizer of K in G. > > I am trying to look at the behaviour of elements of DC under the right > action of N_G(K), and the left action of N_G(H). These actions are > obviously only defined up to elements of the respective underlying > subgroups; but even with the full normalizer I have been unable to figure > out a "natural" way (other than by element-by-"element" multiplication > using the single cosets) to create these actions. > > Is this action something that exists somewhere that I haven't been able to > dig out? You can of course use GAP's mechanism for comparing double cosets and then simply act on double cosets. for example: gap> G:=SymmetricGroup(8); Sym( [ 1 .. 8 ] ) gap> H:=Group((1,2,3),(2,3,4)); Group([ (1,2,3), (2,3,4) ]) gap> K:=Group((5,6,7,8)); Group([ (5,6,7,8) ]) gap> NoH:=Normalizer(G,H); Group([ (2,4,3), (1,2)(3,4), (1,4)(2,3), (2,3), (2,3,4)(6,7), (2,3,4)(6,8), (2,3,4)(5,6,8) ]) gap> NoK:=Normalizer(G,K); Group([ (5,6,7,8), (5,7)(6,8), (2,3), (2,4), (1,2,4), (2,4)(6,8) ]) gap> dc:=DoubleCosets(G,H,K);; gap> Length(dc); 852 Now define an action of NoK on the double cosets by right multiplication right:=function(d,n) return DoubleCoset(H,Representative(d)*n,K);end; Now you can act: gap> Action(NoK,dc,right); <permutation group with 6 generators> gap> List(Orbits(NoK,dc,right),Length); [ 4, 16, 16, 48, 16, 16, 4, 16, 48, 16, 24, 48, 48, 24, 16, 16, 4, 24, 16, 16, 24, 48, 48, 24, 16, 48, 16, 48, 48, 24, 48, 24 ] The cost here is basically comparison of double cosets, which is done by looking at the right cosets involved in each. Clearly this has a limit, but certainly won't be worse than the naive approach. A left action of NoH is easily defined the same way. Hope this is of help, Alexander Hulpke -- Colorado State University, Department of Mathematics, Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA email: hul...@math.colostate.edu, Phone: ++1-970-4914288 http://www.math.colostate.edu/~hulpke _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
