# 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

## Definition

### Symbol-free definition

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

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

## Metaproperties

### Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H \le K \le G$ and $H$ is 3-subnormal in $G$, $H$ is 3-subnormal in $G$. 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 $H ,K \le G$ with $H$ a 3-subnormal subgroup of $G$, $H \cap K$ is 3-subnormal in $K$. 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-closed
ABOUT 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