Group satisfying ascending chain condition on characteristic subgroups

Definition

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

1. Any ascending chain of characteristic subgroups stabilizes after a finite length.
2. Any nonempty collection of characteristic 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.
3. Any characteristic subgroup of the group occurs as the characteristic closure of a finitely generated subgroup, or equivalently, as the characteristic closure of a finite subset.

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
Noetherian group				|FULL LIST, MORE INFO
group satisfying ascending chain condition on subnormal subgroups				|FULL LIST, MORE INFO
group satisfying ascending chain condition on normal subgroups				|FULL LIST, MORE INFO