Using Cayley's Theorem to find an isomorphic subgroup. Find a subgroup of $S_4$ isomorphic to $\mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 2\mathbb{Z}$.
Here is what I have so far: We need to find a subgroup of $S_4$ that is isomorphic to $\{0, 1 \} \times \{0, 1 \}$. Now, $\{0, 1 \} \times \{0, 1 \} = \{(0,0), (0,1), (1,0), (1,1) \}$. The pairs $(0,0)$ and $(1,1)$ tell us that the identity map, $e$, is present. The pairs $(0,1)$ and $(1,0)$ tell us that the transposition map, $\tau$ is present. So a subgroup of $S_4$ isomorphic to $\mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 2\mathbb{Z}$ must be $\{e, \tau \}$.
Comment: This seemed too easy and I'm thinking there must be more. Thank you for your help!

The number $4$ is just large enough to pick two disjoint transpositions. Say
$$
\sigma=(1,2)\qquad\tau=(3,4).
$$
Now check that the subgroup generated by $\sigma$ and $\tau$ does what you want. More generally, you see why $S_{2n}$ contains a subgroup isomorphic to $\mathbb{Z}_2^n$.
