Definition:Sylow p-Subgroup

Definition
Let $p$ be prime.

Let $G$ be a finite group such that $\left|{G}\right| = k p^n$ where $p \nmid k$.

A Sylow $p$-subgroup is a $p$-subgroup of $G$ which has $p^n$ elements.

Alternatively, a Sylow $p$-subgroup of a $G$ is a maximal $p$-subgroup $P$ of $G$.

In this context, maximality means that if $Q$ is a $p$-subgroup of $G$ and $P \le Q$, then $P = Q$.

Thus the divisor $p^n$ which is the largest power of $p$ which divides the order of $G$ is called the maximal prime power divisor corresponding to $p$.

Sylow $p$-subgroups are sometimes called $p$-Sylow subgroups, or just Sylow subgroups.