$p$-group PGroup4 2004-12-12 10:56:47 2007-05-21 04:43:43 Definition p-subgroup primary component p-primary p-primary subgroup primary subgroup \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{component} \PMlinkescapeword{maximal} \PMlinkescapeword{subgroup} \section*{Primary groups} Let $p$ be a prime number. A \emph{$p$-group} (or \emph{$p$-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 \emph{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 $p$. \section*{Primary subgroups} A \emph{$p$-subgroup} (or \emph{$p$-primary subgroup}) of a group $G$ is a \PMlinkname{subgroup}{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 \emph{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 \emph{$p$-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 \PMlinkname{Sylow $p$-subgroups}{SylowPSubgroups}.