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.

Maximal
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.