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'$p$-group'
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| Title of object: |
$p$-group |
| Canonical Name: |
PGroup4 |
| Type: |
Definition |
| Created on: |
2004-12-12 10:56:47 |
| Modified on: |
2005-11-25 17:36:03 |
| Classification: |
msc:20F50 |
| Defines: |
p-subgroup, primary component |
| Synonyms: |
$p$-group=p-group |
Revision comment (for changes between this and next version):
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts} |
Content:
\PMlinkescapeword{maximal}
\PMlinkescapeword{subgroup}
Let $p$ be a prime number.
A \emph{$p$-group} is a group in which the order of every element is a power of $p$.
A \PMlinkname{subgroup}{Subgroup} that is itself a $p$-group is called a \emph{$p$-subgroup}.
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 $p$.
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}. |
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