$S_4$ Isomorphism

Question

I am trying to show that $S_4 \cong X \rtimes_{\varphi} S_3$ where $X= \mathbb{Z}_2 \times \mathbb{Z}_2$. I know that the group of automorphism on $X$ are isomorphic to $S_3$. I'm not sure if that is something I can use to here. Any help would be appreciated.

Answer 1

In order to prove this we can show that $S_4$ is an internal semidirect product of two subgroups.

First we must find a subgroup $N$ of $S_4$ that is normal and isomorfic to the klein $4$ group. In order to find it we notice it must only have elements of order $2$ or $1$ but cannot contain transpositions (because if it has one transposition it must have all of them and then it must be the whole group).

We find that the normal subgroup we need is $\{e,(12)(34),(13)(24),(14)(23)\}$

The next thing we must do is find a subgroup $H$ of $S_4$ that is isomorphic to $S_3$ such that $H\cap N = \{e\}$ and $G=NH$.

In order to do this we consider the most obvious candidate: the permutations that fix $1$, clearly the intersection is trivial because no double cycle can move only $3$ elements. In order to see $S_4=NH$ we use $|H_1 H_2| = |H_1||H_2|/|H_1\cap H_2|$.