[GAP Forum] Action of a group on its double cosets

[Alexander Hulpke]

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: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke