Let $G$ be a finite nonidentity [simple group](/page/Simple%20Group), let $p$ be a [prime number](/page/Prime%20Number) such that $p \mid |G|$, and let $P \le G$ be a Sylow $p$-subgroup. If $P \trianglelefteq G$, then $P=G$. Consequently, if $G$ is a finite non-abelian simple group, then for every prime number $p$ dividing $|G|$, no Sylow $p$-subgroup of $G$ is normal.