$p$-subgroup

Let $G$ be a finite group with order $n$, and let $p$ be a prime integer.We can write $n=p^{k}m$ for some $k,m$ integers, such that $k$ and $m$ are coprimes (that is, $p^k$ is the highest power of $p$ that divides $n$).Any subgroup of $G$ whose order is $p^k$ is called a Sylow $p$-subgroup.

While there is no reason for Sylow $p$-subgroups to exist for any finite group, the fact is that all groups have Sylow $p$-subgroups for every prime $p$ that divides $|G|$. This statement is the First Sylow theorem

When $|G|=p^k$ we simply say that $G$ is a $p$-group.