Show that $S_{4} \cong V \rtimes_{\phi} S_{3}$

Question

The Klein Four-Group is $V = \mathbb{Z}_{2} \times \mathbb{Z}_{2}.$

$(a)$ Show that $Aut(V) \cong S_{3}.$

$(b)$ Use part $(a)$ to show that $S_{4} \cong V \rtimes_{\phi} S_{3}$

My question is:

How to use part $(a)$ to show part $(b)$?

Answer 1

For a): the group $V$ can be identified with the 2-dim vector space over the field of order $2$. So the automorphism group is isomorphic to $GL(2,2)$ which is of order $6$ and non-commutative. So the automorphism group is isomorphic to $S_3$ (the only noncommutative group of order $6$).

b) Let $V$ consist of the identity and all double involutions $(1a)(bc)$ from $S_4$ where $\{a,b,c\} = \{2, 3, 4\}$, $b<c$. And $S$ be the subgroup consisting of all elements fixing $1$. Then $S$ is isomorphic to $S_3$, $S\cap V$ is the identity and $SV= S_4$ by the product formula. So $S_4$ is a semidirect product of $V$ and $S_3$.