This article is about a definition in group theory that is standard among the group theory community (or subcommunity that dabbles in such things) but is not very basic or common for people outside.
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Definition
A group is said to satisfy the subnormal join property if it satisfies the following equivalent conditions:
 The join (i.e., subgroup generated) of two Subnormal subgroup (?)s of the group is again subnormal.
 The join of a finite collection of subnormal subgroups of the group is again subnormal.
 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 


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nilpotent group 
has finite nilpotency class 


Group satisfying generalized subnormal join property, Group with nilpotent derived subgroupFULL LIST, MORE INFO

group in which every subgroup is subnormal 
every subgroup is a subnormal subgroup 


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simple group 
nontrivial; no proper nontrivial normal subgroup 


Group satisfying ascending chain condition on subnormal subgroupsFULL LIST, MORE INFO

Tgroup 
every subnormal subgroup is normal 


Group in which every subnormal subgroup is 2subnormalFULL LIST, MORE INFO

group in which every subnormal subgroup is 2subnormal 
every subnormal subgroup is a 2subnormal subgroup 


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group with nilpotent derived subgroup 
the derived subgroup is a nilpotent group 
nilpotent derived subgroup implies subnormal join property 

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supersolvable group 
has a normal series where all successive quotients are cyclic groups 


Group with nilpotent derived subgroup, Noetherian groupFULL LIST, MORE INFO

finite group 
has finite order 
finite implies subnormal join property 

FZgroup, Group satisfying generalized subnormal join property, Noetherian groupFULL 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 propertyFULL 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 propertyFULL 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 

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