Group in which every subnormal subgroup is 2-subnormal

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 in which every subnormal subgroup is 2-subnormal is a group satisfying the following equivalent conditions:

1. Every subnormal subgroup is a 2-subnormal subgroup, i.e., its subnormal depth (or subnormal defect) is at most $2$.
2. Every 3-subnormal subgroup is a 2-subnormal subgroup.
3. Fix $k>2$. Then, any $k$-subnormal subgroup is a 2-subnormal subgroup.

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (subnormal subgroup) satisfies the second property (2-subnormal subgroup), and vice versa.
View other group properties obtained in this way

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
T-group	every subnormal subgroup is normal			|FULL LIST, MORE INFO
Abelian group	any two elements commute			|FULL LIST, MORE INFO
Dedekind group	every subgroup is normal			|FULL LIST, MORE INFO
Group of nilpotency class two				|FULL LIST, MORE INFO
Group in which every subgroup is 2-subnormal	every subgroup is a 2-subnormal subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group satisfying subnormal join property		2-subnormal implies join-transitively subnormal		|FULL LIST, MORE INFO