Sylow $p\dash$subgroup · Qual Corpus

Let $G$ be a finite group and $p$ a prime, and write $\abs G = p^a m$ with $a\geq 0$ and $p \nmid m$. A subgroup $S \leq G$ is a Sylow $p\dash$subgroup of $G$ if $\abs S = p^a$.

Let $G$ be a finite group and $p$ a prime. A subgroup $S\leq G$ is a Sylow $p\dash$subgroup if and only if $S$ is a $p\dash$group that is maximal under inclusion among the $p\dash$subgroups of $G$.