$p$-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
Canonical name	Psubgroup
Date of creation	2013-03-22 14:02:14
Last modified on	2013-03-22 14:02:14
Owner	drini (3)
Last modified by	drini (3)
Numerical id	8
Author	drini (3)
Entry type	Definition
Classification	msc 20D20
Related topic	PGroup4
Defines	Sylow $p$-subgroup
Defines	Sylow p-subgroup
Defines	$p$-group
Defines	p-group