# 3-subnormal subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and 2-subnormal subgroup

View other such compositions|View all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **3-subnormal subgroup** if it satisfies the following equivalent conditions:

- It is a subnormal subgroup and its subnormal depth is at most three.
- It is a 2-subnormal subgroup of a normal subgroup.
- It is a 2-subnormal subgroup in its normal closure.
- It is a normal subgroup of a 2-subnormal subgroup.

## Relation with other properties

### Stronger properties

- Normal subgroup
- 2-subnormal subgroup
- Commutator of a 2-subnormal subgroup and a subset:
`For full proof, refer: Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal`

### Weaker properties

- Subnormal subgroup
- Conjugate-join-closed subnormal subgroup:
`For full proof, refer: 3-subnormal implies finite-conjugate-join-closed subnormal`

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

A 3-subnormal subgroup of a 3-subnormal subgroup need not be 3-subnormal. This follows from the fact that there can be subgroups of arbitrarily large subnormal depth. `For full proof, refer: There exist subgroups of arbitrarily large subnormal depth`

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If and is 3-subnormal in , is 3-subnormal in . In fact, an analogous statement holds for all subnormal depths. `For full proof, refer: Subnormality satisfies intermediate subgroup condition`

### Transfer condition

YES:This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.

View other subgroup properties satisfying the transfer condition

If with a 3-subnormal subgroup of , is 3-subnormal in . In fact, an analogous statement holds for all subnormal depths. `For full proof, refer: Subnormality satisfies transfer condition`

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of 3-subnormal subgroups is 3-subnormal. An analogous statement holds for all subnormal depths. `For full proof, refer: Subnormality of fixed depth is strongly intersection-closed`

### Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.

Read an article on methods to prove that a subgroup property is not join-closed

A join of two 3-subnormal subgroups need not be 3-subnormal; in fact, it need not even be subnormal. `For full proof, refer: Join of 3-subnormal subgroups need not be subnormal`