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.