# Artinian group

From Groupprops

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

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

## Contents

## Definition

A group is said to be **Artinian** or to satisfy the **minimum condition** on subgroups if every descending chain of subgroups stabilizes after a finite stage.

## Relation with other properties

### Stronger properties

### Weaker properties

- Periodic group:
*For proof of the implication, refer Artinian implies periodic and for proof of its strictness (i.e. the reverse implication being false) refer Periodic not implies Artinian*. - Co-Hopfian group:
*For proof of the implication, refer Artinian implies co-Hopfian and for proof of its strictness (i.e. the reverse implication being false) refer Co-Hopfian not implies Artinian*. - Minimax group