Group satisfying ascending chain condition on normal subgroups

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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:

  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.

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 Group of finite chief length, Group of finite composition length, Group of finite max-length, Noetherian group|FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated Group satisfying ascending chain condition on subnormal subgroups|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 Group of finite chief length, Group satisfying ascending chain condition on subnormal subgroups|FULL LIST, MORE INFO
simple group no proper nontrivial normal subgroup Group of finite chief length, Group satisfying ascending chain condition on subnormal subgroups|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