Transitively normal subgroup

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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]

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This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

CAUTIONARY NOTE: There is a paper where the term transitively normal is used for what we call intermediately subnormal-to-normal subgroup

Definition

Symbol-free definition

A subgroup of a group is termed transitively normal or right-transitively normal if every normal subgroup of it is a normal subgroup of the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed transitively normal if whenever $K$ is a normal subgroup of $H$, $K$ is also a normal subgroup of $G$.

Formalisms

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Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression	$H$ is a transitively normal subgroup of $G$ if ...	This means that transitive normality is ...	Additional comments
normal automorphism $\to$ normal automorphism	every normal automorphism of $G$ restricts to a normal automorphism of $H$	the balanced subgroup property for normal automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism $\to$ normal automorphism	every inner automorphism of $G$ restricts to a normal automorphism of $H$
class-preserving automorphism $\to$ normal automorphism	every class-preserving automorphism of $G$ restricts to a normal automorphism of $H$
subgroup-conjugating automorphism $\to$ normal automorphism	every subgroup-conjugating automorphism of $G$ restricts to a normal automorphism of $H$
strong monomial automorphism $\to$ normal automorphism	every strong monomial automorphism of $G$ restricts to a normal automorphism of $H$

In terms of the right transiter

This property is obtained by applying the right transiter to the property: normal subgroup
View other properties obtained by applying the right transiter

Relation with other properties

Conjunction with other properties

Conjunctions with subgroup properties:

• Characteristic transitively normal subgroup is a transitively normal subgroup that is also a characteristic subgroup.
• c-closed transitively normal subgroup is a transitively normal subgroup that is also a c-closed subgroup.

Conjunctions with group properties:

• Nilpotent transitively normal subgroup is a transitively normal subgroup that is also a nilpotent group.

Stronger properties

• Direct factor
• Central factor
• Central subgroup
• Cocentral subgroup
• Conjugacy-closed normal subgroup: A subgroup that is both conjugacy-closed and normal
• SCAB-subgroup
• Hereditarily normal subgroup

Weaker properties

• Normal subgroup
• CEP-subgroup
• Subgroup whose commutator with any subset is normal: For full proof, refer: Commutator of a transitively normal subgroup and a subset implies normal

Metaproperties

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Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this 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 transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

This follows on account of its being a balanced subgroup property.

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 transitively normal in $G$, $H$ is also transitively normal in any intermediate subgroup of $G$ containing it. This follows from the general fact that normality satisfies intermediate subgroup condition, and the right transiter of any property satisfying intermeditae subgroup condition also satisfies intermediate subgroup condition.

For full proof, refer: Transitive normality satisfies intermediate subgroup condition, Normality satisfies intermediate subgroup condition, Intermediate subgroup condition is right residual-preserved

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of two transitively normal subgroups need not be transitively normal. This follows from the direct product technique. In fact, we can consider normal subgroups of index two that are not transitively normal (for instance, the power of the base in a wreath product by $\mathbb{Z}/2\mathbb{Z}$).

For full proof, refer: Transitive normality is not finite-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 transitively normal subgroups need not be transitively normal.

For full proof, refer: Transitive normality is not finite-join-closed

Centralizer-closedness

This subgroup property is not centralizer-closed: the centralizer of any subgroup with this property, in the whole group, need not have this property.

The centralizer of a transitively normal subgroup need not be transitively normal.

For full proof, refer: Transitive normality is not centralizer-closed

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

The image of a transitively normal subgroup under a surjective homomorphism is transitively normal in the image. For full proof, refer: Transitive normality satisfies image condition

Quotient-transitivity

This subgroup property is not quotient-transitive: the corresponding quotient property is transitive.

If $H\le K\le G$ are groups such that $H$ is transitively normal in $K$ and $K/H$ is transitively normal in $G/H$, it is not necessary that $K$ is transitively normal in $G$.

For full proof, refer: Transitive normality is not quotient-transitive