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 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 |
