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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.

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  • $\begingroup$ I think this can help.. $\endgroup$ – Bman72 Dec 12 '14 at 17:24
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    $\begingroup$ 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. $\endgroup$ – Alex Wertheim Dec 12 '14 at 17:25

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