# $p$-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.

Title $p$-subgroup Psubgroup 2013-03-22 14:02:14 2013-03-22 14:02:14 drini (3) drini (3) 8 drini (3) Definition msc 20D20 PGroup4 Sylow $p$-subgroup Sylow p-subgroup $p$-group p-group