minimal condition

A group is said to satisfy the minimal condition if every strictly descending chain of subgroupsMathworldPlanetmathPlanetmath


is finite.

This is also called the descending chain conditionMathworldPlanetmathPlanetmath.

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


is an infinite descending chain of subgroups.

Similar properties are useful in other classes of algebraic structuresPlanetmathPlanetmath: see for example the ArtinianPlanetmathPlanetmath 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