# 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 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.
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 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.

## 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

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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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.
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 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 $G$.