[GAP Forum] permutation actions

[Alexander Hulpke]

Dear GAP-Forum,

Anton Konygin wrote:

> Let G be a group generated by permutations g_1, ..., g_d on the set  
> {1,
> ..., n} and H be a subgroup of G generated by permutations  
> h_1, ..., h_k
> on the set {1, ..., n} (g_1, ..., g_d, h_1, ..., h_k are given
> explicitly).
>
> Can I use GAP to get the natural (permutation) action of g_1, ...,  
> g_d on
> left cosets of G by H (ordering in some way)?

GAP uses right cosets and action from the right:

A:=Action(G,RightTransversal(G,H),OnRight);

Then
GeneratorsOfGroup(A) give the action of g1,..g_d.
(To get the (left) action on the left cosets, you would have to  
simply invert the permutations.)

If you want the homomorphism, you can use
phi:=ActionHomomorphism(G,RightTransversal(G,H),OnRight);

Best wishes,

     Alexander Hulpke