# maximal condition

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

 $G_{1}\subset G_{2}\subset G_{3}\subset\cdots$

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 MaximalCondition 2013-03-22 13:58:47 2013-03-22 13:58:47 mclase (549) mclase (549) 6 mclase (549) Definition msc 20D30 ascending chain condition