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