Cayley's Representation Theorem/Proof 1

Theorem
Let $S_n$ denote the symmetric group on $n$ letters.

Every finite group is isomorphic to a subgroup of $S_n$ for some $n \in \Z$.

Proof
Let $H = \left\{{e}\right\}$.

We can apply Permutation of Cosets to $H$ so that $\mathbb S = G$ and $\ker \left({\theta}\right) = \left\{{e}\right\}$.

The result follows by the First Isomorphism Theorem.