# $p$-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 |