Group satisfying ascending chain condition on subnormal 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

Symbol-free definition

A group is said to satisfy the ascending chain condition on subnormal subgroups or the maximum condition on subnormal subgroups if it satisfies the following equivalent conditions:

  1. Any ascending chain of subnormal subgroups stabilizes after a finite length.
  2. Any nonempty collection of subnormal subgroups has a maximal element: a member that is not contained in any other member of the collection.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Noetherian group |FULL LIST, MORE INFO
group of finite composition length |FULL LIST, MORE INFO
simple group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group satisfying subnormal join property ascending chain condition on subnormal subgroups implies subnormal join property Group satisfying generalized subnormal join property|FULL LIST, MORE INFO
group satisfying generalized subnormal join property ascending chain condition on subnormal subgroups implies generalized subnormal join property |FULL LIST, MORE INFO
group satisfying ascending chain condition on normal subgroups |FULL LIST, MORE INFO
group satisfying ascending chain condition on characteristic subgroups Group satisfying ascending chain condition on normal subgroups|FULL LIST, MORE INFO
Hopfian group Group satisfying ascending chain condition on normal subgroups|FULL LIST, MORE INFO