# Group of finite max-length

From Groupprops

## Contents

## Definition

A **group of finite max-length** is a group satisfying the following equivalent conditions:

- The group is both a Noetherian group (i.e., it satisfies the ascending chain condition on all subgroups) and an Artinian group (i.e.,it satisfies the descending chain condition on all subgroups).
- Given any chain of subgroups of finite length, there is a chain of subgroups of finite length that refines it and that cannot be refined further.
- The max-length of the group is finite.

A group of finite max-length need not be finite, though counterexamples are rare. The best counterexamples are Tarski monsters, which have max-length two but are infinite.

### Equivalence of definitions

The equivalence of definition relies on a Konig's lemma-style idea.

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: Noetherian group and Artinian group

View other group property conjunctions OR view all group properties

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Noetherian group | ||||

Artinian group | ||||

group of finite chief length | ||||

group of finite composition length | ||||

group satisfying ascending chain condition on normal subgroups | ||||

group satisfying descending chain condition on normal subgroups | ||||

group satisfying ascending chain condition on subnormal subgroups | ||||

group satisfying descending chain condition on subnormal subgroups |