Consider the following subgroup of the symmetric group $S_4$:
$$V_4 = \{(1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}.$$
(a). Show that $V_4$ is a normal subgroup of $S_4$.
(b). Find a permutation $\alpha \in A_4$ such that the three cosets of $V_4$ in $A_4$ are $V_4$, $\alpha V_4$ and $\alpha^2V_4.$
For question a), would I have to show that the elements of $V_4$ are isomorphic to the elements in $S_4$? Or would it involve the use of cosets, since the definition of a normal subgroup is $xH = Hx$? (Left and right cosets coincide.)
As for b), I'm really not sure where to begin.