Group satisfying subnormal join property

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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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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 is said to satisfy the subnormal join property if it satisfies the following equivalent conditions:

  1. The join (i.e., subgroup generated) of two Subnormal subgroup (?)s of the group is again subnormal.
  2. The join of a finite collection of subnormal subgroups of the group is again subnormal.
  3. The commutator of any two subnormal subgroups of the group is again subnormal.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group satisfying generalized subnormal join property an arbitrary join of subnormal subgroups is subnormal |FULL LIST, MORE INFO
nilpotent group has finite nilpotency class Group satisfying generalized subnormal join property, Group with nilpotent derived subgroup|FULL LIST, MORE INFO
group in which every subgroup is subnormal every subgroup is a subnormal subgroup |FULL LIST, MORE INFO
simple group nontrivial; no proper nontrivial normal subgroup Group satisfying ascending chain condition on subnormal subgroups|FULL LIST, MORE INFO
T-group every subnormal subgroup is normal Group in which every subnormal subgroup is 2-subnormal|FULL LIST, MORE INFO
group in which every subnormal subgroup is 2-subnormal every subnormal subgroup is a 2-subnormal subgroup |FULL LIST, MORE INFO
group with nilpotent derived subgroup the derived subgroup is a nilpotent group nilpotent derived subgroup implies subnormal join property |FULL LIST, MORE INFO
supersolvable group has a normal series where all successive quotients are cyclic groups Group with nilpotent derived subgroup, Noetherian group|FULL LIST, MORE INFO
finite group has finite order finite implies subnormal join property FZ-group, Group satisfying generalized subnormal join property, Noetherian group|FULL LIST, MORE INFO
Noetherian group (also called slender group) every subgroup is finitely generated Noetherian implies subnormal join property Group satisfying ascending chain condition on subnormal subgroups, Group satisfying generalized subnormal join property|FULL LIST, MORE INFO
group satisfying ascending chain condition on subnormal subgroups no infinite strictly ascending chain of subnormal subgroups ascending chain condition on subnormal subgroups implies subnormal join property Group satisfying generalized subnormal join property|FULL LIST, MORE INFO
group whose derived subgroup satisfies ascending chain condition on subnormal subgroups derived subgroup contains no infinite strictly ascending chain of subnormal subgroups derived subgroup satisfies ascending chain condition on subnormal subgroups implies subnormal join property |FULL LIST, MORE INFO

References

Textbook references