# $p$-group

## Primary groups

Let $p$ be a prime number. A $p$-group (or $p$-primary group) is a group in which the order of every element is a power of $p$. A group that is a $p$-group for some prime $p$ is also called a primary group.

Using Lagrange’s Theorem and Cauchy’s Theorem one may show that a finite group $G$ is a $p$-group if and only if $|G|$ is a power of $p$.

## Primary subgroups

A $p$-subgroup (or $p$-primary subgroup) of a group $G$ is a subgroup (http://planetmath.org/Subgroup) $H$ of $G$ such that $H$ is also a $p$-group. A group that is a $p$-subgroup for some prime $p$ is also called a primary subgroup.

It follows from Zorn’s Lemma that every group has a maximal $p$-subgroup, for every prime $p$. The maximal $p$-subgroup need not be unique (though for abelian groups it is always unique, and is called the $p$-primary component of the abelian group). A maximal $p$-subgroup may, of course, be trivial. Non-trivial maximal $p$-subgroups of finite groups are called Sylow $p$-subgroups (http://planetmath.org/SylowPSubgroups).

 Title $p$-group Canonical name Pgroup Date of creation 2013-03-22 14:53:08 Last modified on 2013-03-22 14:53:08 Owner yark (2760) Last modified by yark (2760) Numerical id 13 Author yark (2760) Entry type Definition Classification msc 20F50 Synonym p-group Synonym p-primary group Synonym primary group Related topic PGroup Related topic PExtension Related topic ProPGroup Related topic QuasicyclicGroup Related topic Subgroup Defines p-subgroup Defines primary component Defines p-primary Defines p-primary subgroup Defines primary subgroup