Conjugacy-closed normal subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: conjugacy-closed subgroup and normal subgroup
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Definition

Symbol-free definition

A subgroup of a group is termed conjugacy-closed normal if it satisfies the following equivalent conditions:

It is normal as well as conjugacy-closed (viz any two elements of the subgroup that are conjugate in the whole group are also conjugate in the subgroup).

Every class automorphism of the whole group restricts to a class automorphism of the subgroup (a class automorphism is an automorphism that sends each element to within its conjugacy class).

Every inner automorphism of the whole group restricts to a class automorphism of the subgroup.

Definition with symbols

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

$H\triangleleft G$ and whenever $gxg^{-1}=y$ for $x,y\in H$ and $g\in G$ , there exists $h\in H$ such that $hxh^{-1}=y$ .

Given any automorphism $\sigma$ of $G$ such that for every $x\in G$ , there exists $g\in G$ such that $\sigma (x)=gxg^{-1}$ , we also have the following: for every $x\in H$ there exists $h\in H$ such that $\sigma (x)=hxh^{-1}$ .

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

Relation with other properties

Stronger properties

Weaker properties

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.
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if $H$ is conjugacy-closed normal in $K$ and $K$ is conjugacy-closed normal in $G$ , then $H$ is conjugacy-closed normal in $G$ . This follows directly from conjugacy-closed normality being a balanced subgroup property in the function restriction formalism.

Intersection-closedness

It is not clear whether the intersection of two conjugacy-closed normal subgroups is conjugacy-closed normal.

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).
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Clearly, every group is conjugacy-closed normal as a subgroup of itself. Further, the trivial subgroup is also clearly conjugacy-closed normal in the whole group.

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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If $H$ is conjugacy-closed normal in $G$ , it is also conjugacy-closed normal in every intermediate subgroup $K$ . This follows from these two facts:

If $H$ is conjugacy-closed in $G$ , $H$ is conjugacy-closed in any intermediate subgroup $K$ of $G$ .
If $H$ is normal in $G$ , $H$ is noraml in every intermediate subgroup $K$ of $G$ .