Cosets of a Normal Subgroup, Quotient Group

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.

• I think this can help.. – Bman72 Dec 12 '14 at 17:24
• Your approach to (a) doesn't make any sense to me. What do you mean when you say that the elements of $V_{4}$ are isomorphic to the elements of $S_{4}$? The easiest way to do (a) is to use the observation that conjugation preserves cycle type. – Alex Wertheim Dec 12 '14 at 17:25