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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?

Many thanks for any helpful comments or suggestions.

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?

Many thanks for any helpful comments or suggestions.

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