Group satisfying ascending chain condition on normal subgroups
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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
|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|