Group satisfying subnormal join property

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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			|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			|FULL LIST, MORE INFO
T-group	every subnormal subgroup is normal			|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			|FULL LIST, MORE INFO
finite group	has finite order	finite implies subnormal join property		|FULL LIST, MORE INFO
Noetherian group (also called slender group)	every subgroup is finitely generated	Noetherian implies 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		|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

• A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 388 (definition in paragraph)