Transitively normal subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Transitively normal subgroup, all facts related to Transitively normal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 it satisfies the following equivalent conditions:

1. Every normal subgroup of it is a normal subgroup of the whole group.
2. Every normal automorphism of the whole group restricts to a normal automorphism of the subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed transitively normal if it satisfies the following equivalent conditions:

1. Whenever $K$ is a normal subgroup of $H$, $K$ is also a normal subgroup of $G$.
2. For any normal automorphism $\sigma$ of $G$, the restriction of $\sigma$ to $H$ is a normal automorphism of $H$.

Equivalence of definitions

Further information: Equivalence of definitions of transitively normal subgroup

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Central factor	product with centralizer is whole group	central factor implies transitively normal	transitively normal not implies central factor (see also list of examples)	|FULL LIST, MORE INFO
Direct factor	factor in an internal direct product	direct factor implies transitively normal (also, via central factor)	(via central factor) (see also list of examples)	|FULL LIST, MORE INFO
Cocentral subgroup	product with center is whole group]]	(via central factor)	(via central factor)	|FULL LIST, MORE INFO
Central subgroup	contained in the center	(via central factor, via hereditarily normal)	(via central factor, via hereditarily normal)	|FULL LIST, MORE INFO
Conjugacy-closed normal subgroup	both conjugacy-closed and normal	conjugacy-closed normal implies transitively normal	transitively normal not implies conjugacy-closed	|FULL LIST, MORE INFO
SCAB-subgroup	any subgroup-conjugating automorphism of whole group restricts to subgroup-conjugating automorphism of subgroup	SCAB implies transitively normal	Transitively normal not implies SCAB	|FULL LIST, MORE INFO
Hereditarily normal subgroup	every subgroup of it is normal in whole group			|FULL LIST, MORE INFO
Cyclic normal subgroup	cyclic and normal	(via SCAB, via hereditarily normal)	(via SCAB, via hereditarily normal)	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup		(by definition)	normality is not transitive	|FULL LIST, MORE INFO
CEP-subgroup	every normal subgroup of it arises as its intersection with a normal subgroup of whole group			|FULL LIST, MORE INFO
Subgroup whose commutator with any subset is normal	its commutator with any subset of whole group is normal in whole group	commutator of a transitively normal subgroup and a subset implies normal	commutator with any subset is normal not implies transitively normal	|FULL LIST, MORE INFO

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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