# Group of finite max-length

## Definition

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

1. 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).
2. 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.
3. 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 properties
VIEW 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