# 4-subnormal subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a 4-subnormal subgroup if it satisfies the following equivalent conditions:

## 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 4-subnormal subgroup of a 4-subnormal subgroup need not be 4-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 4-subnormal in $G$, $H$ is 4-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 4-subnormal subgroup of $G$, $H \cap K$ is 4-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 4-subnormal subgroups is 4-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 4-subnormal subgroups need not be 4-subnormal; in fact, it need not even be subnormal. More generally, a join of 3-subnormal subgroups need not be subnormal. For full proof, refer: Join of 3-subnormal subgroups need not be subnormal