Group satisfying ascending chain condition on subnormal 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
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:
- Any ascending chain of subnormal subgroups stabilizes after a finite length.
- 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 |