# maximal subgroup

Let $G$ be a group.

A subgroup^{} $H$ of $G$
is said to be a *maximal subgroup* of $G$
if $H\ne G$ and there is no subgroup $K$ of $G$
such that $$.
Note that a maximal subgroup of $G$ is not maximal (http://planetmath.org/MaximalElement) among all subgroups of $G$,
but only among all proper subgroups^{} of $G$.
For this reason, maximal subgroups are sometimes called *maximal proper subgroups*.

Similarly, a normal subgroup^{} $N$ of $G$
is said to be a *maximal normal subgroup*
(or *maximal proper normal subgroup*) of $G$
if $N\ne G$ and there is no normal subgroup $K$ of $G$
such that $$.
We have the following theorem:

###### Theorem.

A normal subgroup $N$ of a group $G$ is a maximal normal subgroup if and only if the quotient (http://planetmath.org/QuotientGroup) $G\mathrm{/}N$ is simple (http://planetmath.org/Simple).

Title | maximal subgroup |

Canonical name | MaximalSubgroup |

Date of creation | 2013-03-22 12:23:46 |

Last modified on | 2013-03-22 12:23:46 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 15 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E28 |

Synonym | maximal proper subgroup |

Related topic | MaximalElement |

Defines | maximal |

Defines | maximal normal subgroup |

Defines | maximal proper normal subgroup |

Defines | simplicity of quotient group |