PlanetMath: maximal condition

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maximal condition

(Definition)

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

$\displaystyle G_1 \subset G_2 \subset G_3 \subset \cdots$

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.

"maximal condition" is owned by mclase.

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Other names:

ascending chain condition

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Cross-references: modules, rings, Noetherian, algebraic structures, classes, properties, similar, finitely generated, finite, subgroups, chain, strictly, group
There are 5 references to this entry.

This is version 3 of maximal condition, born on 2003-10-04, modified 2006-09-12.
Object id is 4752, canonical name is MaximalCondition.
Accessed 3579 times total.

Classification:

20D30 (Group theory and generalizations :: Abstract finite groups :: Series and lattices of subgroups)

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