Cayley's Representation Theorem

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

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

Corollary
Let $$G$$ be a group and let $$p$$ be the smallest prime such that $$p \backslash \left|{G}\right|$$.

If $$\exists H: H \le G$$ such that $$\left|{H}\right| = p$$, then $$H \triangleleft G$$.

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

The result follows by the First Isomorphism Theorem.

Proof of Corollary
Apply Permutation of Cosets: Corollary to $$H$$ to find some $$N \triangleleft G$$ such that $$\left[{G : N}\right] \backslash p!$$.

Since $$\left[{G : N}\right] \backslash \left|{G}\right|$$, it divides $$\gcd \left\{{\left|{G}\right|, p!}\right\}$$.

Since $$p$$ is the smallest prime dividing $$\left|{G}\right|$$, $$\gcd \left\{{\left|{G}\right|, p!}\right\} = p$$.

Thus $$\left[{G : N}\right] = p = \left[{G : H}\right]$$.

Since $$N \subseteq H$$, it must follow that $$N = H$$.