2-subnormal subgroup

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and normal subgroup
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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

Definition

QUICK PHRASES: normal inside normal closure, every conjugate is in its normalizer, normal closure is in normalizer, normal subgroup of normal subgroup

Symbol-free definition

A subgroup of a group is termed 2-subnormal if the following equivalent conditions hold:

There is an intermediate subgroup containing it such that the subgroup is normal in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group.
The subgroup is normal in its normal closure.
The normal closure of the subgroup is contained in its normalizer
The subgroup is contained in the normal core of its normalizer

The property of being 2-subnormal is the same as the property of being subnormal of depth 2.

Definition with symbols

A subgroup $H$ of a group $G$ is termed 2-subnormal if the following equivalent conditions hold:

There is subgroup $K$ such that $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$ .
The normal closure of $H$ is a normal subgroup of $G$ .

Formalisms

First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.
View a complete list of first-order subgroup properties

A subgroup $H$ is 2-subnormal in a group $G$ if it satisfies the following first-order sentence:

$\forall g \in G, \forall x, y \in H, g x g^{- 1} y g x^{- 1} g^{- 1} \in H$

Relation with other properties

Stronger properties

Base of a wreath product
Normal subgroup: This follows directly from the definition. The strictness of the implication follows from the fact that normality is not transitive
2-hypernormalized subgroup: This is a particular case of the fact that any $k$ -hypernormalized subgroup is also $k$ -subnormal.
Right-transitively 2-subnormal subgroup
Left-transitively 2-subnormal subgroup
Direct factor of characteristic subgroup
Direct factor of normal subgroup
Normal subgroup of characteristic subgroup
Join-transitively 2-subnormal subgroup
Commutator of a normal subgroup and a subset: Further information: Commutator of a normal subgroup and a subset implies 2-subnormal

Weaker properties

Conjugate-permutable subgroup: For full proof, refer: 2-subnormal implies conjugate-permutable
Subnormal subgroup: This follows directly from the definition.
Join of finitely many 2-subnormal subgroups
Join of 2-subnormal subgroups
3-subnormal subgroup
Join-transitively subnormal subgroup: For full proof, refer: 2-subnormal implies join-transitively subnormal

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 2-subnormal subgroup of a 2-subnormal subgroup is not necessarily 2-subnormal. For full proof, refer: 2-subnormality is not transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Every group is 2-subnormal as a subgroup of itself, and further, the trivial subgroup is 2-subnormal in any group.

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 2-subnormal subgroups is 2-subnormal. For full proof, refer: 2-subnormality 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

An arbitrary join of 2-subnormal subgroups need not be 2-subnormal. In fact, even a join of two 2-subnormal subgroups need not be 2-subnormal. For full proof, refer: 2-subnormality is not finite-join-closed

Conjugate-join-closedness

This subgroup property is conjugate-join-closed; in other words, a join of conjugate subgroups, each having the property, also has the property.
View a complete list of conjugate-join-closed subgroup properties

A join of 2-subnormal subgroups that are conjugate to each other is again 2-subnormal. For full proof, refer: 2-subnormality is conjugate-join-closed

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 $H$ is a 2-subnormal subgroup of $G$ , then $H$ is also 2-subnormal in any intermediate subgroup $K$ of $G$ . For full proof, refer: 2-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 $H$ is a 2-subnormal subgroup of $G$ and $K$ is any subgroup of $G$ , then $H \cap K$ is 2-subnormal in $K$ . For full proof, refer: 2-subnormality satisfies transfer condition

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

If $f : G \to K$ is a surjective homomorphism of groups, and $H$ is 2-subnormal in $G$ , then $f (H)$ is 2-subnormal in $K$ .

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

If $H \leq G$ and $K, L$ are two intermediate subgroups containing $H$ , it may happen that $H$ is 2-subnormal in $K$ as well as in $L$ , but is not 2-subnormal in $⟨ K, L ⟩$ . For full proof, refer: 2-subnormality is not upper join-closed

Effect of property operators

The right transiter

Applying the right transiter to this property gives: right-transitively 2-subnormal subgroup

The right transiter of the property of being 2-subnormal is termed the property of being right-transitively 2-subnormal. A subgroup $H$ of a group $G$ is termed right-transitively 2-subnormal if any 2-subnormal subgroup $K$ of $H$ is 2-subnormal in $G$ .

Some subgroup properties stronger than being right-transitively 2-subnormal include: base of a wreath product, transitively normal subgroup, and normal subgroup that is also a T-group (for instance, an Abelian normal subgroup).

The left transiter

Applying the left transiter to this property gives: left-transitively 2-subnormal subgroup

The left transiter of the property of being 2-subnormal is termed the property of being left-transitively 2-subnormal. A subgroup $H$ of a group $G$ is termed left-transitively 2-subnormal if whenever $G$ is embedded as a 2-subnormal subgroup of some group $K$ , $H$ is also 2-subnormal in $K$ .

Any characteristic subgroup is left-transitively 2-subnormal, because the left transiter of normal is characteristic.