# Group in which every subnormal subgroup is 2-subnormal

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

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

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