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

- and whenever for and , there exists such that .

- Given any automorphism of such that for every , there exists such that , we also have the following: for every there exists such that .

## 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 | is a conjugacy-closed normal subgroup of if ... | This means that conjugacy-closed normality is ... | Additional comments |
---|---|---|---|

class-preserving automorphism class-preserving automorphism | every class-preserving automorphism of restricts to a class-preserving automorphism of | the balanced subgroup property for class-preserving automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |

inner automorphism class-preserving automorphism | every inner automorphism of restricts to a class-preserving automorphism of | a left-inner subgroup property. |

## Relation with other properties

### Stronger properties

### Weaker properties

## 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 transitiveABOUT 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 is conjugacy-closed normal in and is conjugacy-closed normal in , then is conjugacy-closed normal in . 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 conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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

- If is conjugacy-closed in , is conjugacy-closed in any intermediate subgroup of .
- If is normal in , is noraml in every intermediate subgroup of .