# maximal condition

A group is said to satisfy the *maximal condition* if every strictly ascending chain of subgroups^{}

$${G}_{1}\subset {G}_{2}\subset {G}_{3}\subset \mathrm{\cdots}$$ |

is finite.

This is also called the *ascending chain condition ^{}*.

A group satisfies the maximal condition if and only if the group and all its subgroups are finitely generated^{}.

Similar properties are useful in other classes of algebraic structures^{}: see for example the Noetherian^{} condition for rings and modules.

Title | maximal condition |
---|---|

Canonical name | MaximalCondition |

Date of creation | 2013-03-22 13:58:47 |

Last modified on | 2013-03-22 13:58:47 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 6 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20D30 |

Synonym | ascending chain condition |