Sylow normality criterion

Proposition (If $n_p=1$ then the Sylow p-subgroup is normal). Let $G$ be a finite group and let $p$ be a prime dividing $|G|$. Let $P$ be a Sylow p-subgroup of $G$, and let $n_p$ denote the number of Sylow $p$-subgroups of $G$. If $n_p=1$, then $P$ is a normal subgroup of $G$.

Context. Conjugation sends Sylow $p$-subgroups to Sylow $p$-subgroups (and in fact they are all conjugate by Sylow's second theorem ). If there is only one, it must be fixed by conjugation.

Proof sketch. For any $g\in G$, the subgroup $gPg^{-1}$ is again a Sylow $p$-subgroup (it has the same order as $P$). If $n_p=1$, then necessarily $gPg^{-1}=P$ for all $g\in G$. This is exactly the definition of normality: $P\lhd G$.