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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CAUTIONARY NOTE: There is a book 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 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.
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The property of being transitively normal is the balanced subgroup property (function restriction formalism) with respect to the function restriction formalism, with the corresponding function property being the property of being a normal automorphism. In symbols:

Transitively normal subgroup = Normal automorphism → Normal automorphism

Here, quotientable automorphism means an automorhpism that takes every normal subgroup to itself.

The property of being transitively normal arises naturally as the right transiter of the subgroup property of normality.

Relation with other properties

Stronger properties

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

Weaker properties

• Normal subgroup
• CEP-subgroup

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.
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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is transitively normal in $G$, then any normal subgroup $N$ of $H$ is normal in $G$. Now, since normality satisfies the intermediate subgroup condition, $N$ is also normal in every intermediate subgroup $M$ containing $H$. Hence, for any intermediate subgroup $N$ containing $H$, any normal subgroup of $N$ is normal inside that intermediate subgroup.

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 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 intersection-closed