# 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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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