Group satisfying ascending chain condition on normal subgroups

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:


 * 1) Any ascending chain of normal subgroups stabilizes after a finite length.
 * 2) 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.
 * 3) 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.