# Group satisfying ascending chain condition on normal subgroups

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

## Contents

## Definition

A **group satisfying ascending chain condition on normal subgroups** or a **group satisfying maximum condition on normal subgroups** is a group satisfying the following equivalent conditions:

- Any ascending chain of normal subgroups stabilizes after a finite length.
- Any nonempty collection of normal subgroups has a maximal element: in other words, there is a member of that collection that is not contained in any other member of that collection.
- Any normal subgroup of the group occurs as the normal closure of a finitely generated subgroup, or equivalently, as the normal closure of a finite subset.

## Relation with other properties

### Stronger properties

### Weaker properties

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

Hopfian group | every surjective endomorphism is an automorphism | Ascending chain condition on normal subgroups implies Hopfian | |FULL LIST, MORE INFO | |

direct product of finitely many directly indecomposable groups | isomorphic to the external direct product of finitely many directly indecomposable groups | |FULL LIST, MORE INFO | ||

group satisfying ascending chain condition on characteristic subgroups | no infinite strictly ascending chain of characteristic subgroups | |FULL LIST, MORE INFO | ||

group in which any infinite strictly ascending chain of normal subgroups has union the whole group |