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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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
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| finite group | has a finite order | |FULL LIST, MORE INFO | ||
| Noetherian group (also called slender group) | every subgroup is finitely generated | |FULL LIST, MORE INFO | ||
| group satisfying ascending chain condition on subnormal subgroups | no infinite strictly ascending chain of subnormal subgroups | |FULL LIST, MORE INFO | ||
| group of finite chief length | has a chief series of finite length | |FULL LIST, MORE INFO | ||
| group of finite composition length | has a composition series of finite length | |FULL LIST, MORE INFO | ||
| simple group | no proper nontrivial normal subgroup | |FULL LIST, MORE INFO |
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 |