# Group action on cosets of subgroups in non-abelian groups

This is not a homework question, just a general question.

Let G be a non-abelian finite group, S < G a non-normal proper subgroup of index v >= 2, and G/S the set of v right cosets S_1 = S, S_2, ..., S_v, of S in G.

We know there is a naturally defined right-multiplication action G x G/S --> G/S defined by (g,S_i) |--> (S_i)g, and this action is a permutation action on G/S. So for any element g in G, the map \phi_g : G/S --> G/S defined by \phi_g(S_i) = (S_i)g is an element of Sym(v).

If \phi: G --> Sym(v) is the map which sends each g in G to \phi(g), and S < G is a proper subgroup, then what are the conditions for this map to be necessarily surjective? It seems that for v = 3, no additional conditions are required beyond non-normality of the subgroup S. But is this necessarily true for v >= 4?

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Take G the dihedral group of order 8. Let S={1,b} (so the identity, and one reflection). Then S has index 4. Thus Sym(v) has 24 elements. So, since Sym(v) has more elements then G, no surjection G--> Sym(v) can exist...

I want to ask a question first why \phi_g is an element of Sym(v)?why should there is a S_j equals to S_i*g?

thanks