# minimal condition

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

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

is finite.

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

A group which satisfies the minimal condition is necessarily periodic. For if it contained an element $x$ of infinite order, then

$$\u27e8x\u27e9\supset \u27e8{x}^{2}\u27e9\supset \u27e8{x}^{4}\u27e9\supset \mathrm{\cdots}\supset \u27e8{x}^{{2}^{n}}\u27e9\supset \mathrm{\cdots}$$ |

is an infinite descending chain of subgroups.

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

Title | minimal condition |
---|---|

Canonical name | MinimalCondition |

Date of creation | 2013-03-22 13:58:49 |

Last modified on | 2013-03-22 13:58:49 |

Owner | mclase (549) |

Last modified by | mclase (549) |

Numerical id | 4 |

Author | mclase (549) |

Entry type | Definition |

Classification | msc 20D30 |

Synonym | descending chain condition |

Related topic | ChernikovGroup |