(i) The subgroup $V = {(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)}$ is a proper normal subgroup of $A_4$. To see this, note that $V$ is a subgroup of $A_4$ since it is closed under multiplication and inverses. Moreover, it is normal since it is the kernel of the sign homomorphism from $A_4$ to $\mathbb{Z}/2\mathbb{Z}$, which maps even permutations to $0$ and odd permutations to $1$. To see that $V$ is proper, note that it has index $2$ in $A_4$ since $A_4$ has order $12$ and $V$ has order $4$, so $A_4/V$ has order $2$. But there is no subgroup of order $2$ in $A_4$.

The group $V$ is isomorphic to the Klein four-group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

(ii) The left cosets of $V$ inside $A_4$ are $V$ and $\tau V$, where $\tau$ is any transposition not in $V$. For example, if we take $\tau = (1,2)$, then we have

$V={(),(12)(34),(13)(24),(14)(23)}$

and

$\tau V={(12),(12)(34)(12),(13)(24)(12),(14)(23)(12)}$

\begin{prob}

(iii) To which group is $A_4 / V$ isomorphic to?

\end{prob}

\begin{proof}

(iii) The group $A_4/V$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$, the cyclic group of order $2$. To see this, note that $A_4/V$ has order $2$, so it is either isomorphic to $\mathbb{Z}/2\mathbb{Z}$ or to the trivial group ${1}$. But if $A_4/V \cong {1}$, then $A_4 = V$, which is false. Therefore, $A_4/V \cong \mathbb{Z}/2\mathbb{Z}$.