Permutation of Cosets/Corollary 2

Corollary to Permutation of Cosets
Let $G$ be a group and let $p$ be the smallest prime such that:
 * $p \mathop \backslash \left|{G}\right|$

where $\backslash$ denotes Divides.

If $\exists H: H \le G$ such that $\left|{H}\right| = p$, then $H$ is a normal subgroup of $G$.

Proof
Apply Permutation of Cosets: Corollary 1 to $H$ to find some $N \lhd G$ such that:
 * $\left[{G : N}\right] \mathop \backslash p!$

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

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

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

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