# Central factor of normalizer

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]

This is a variation of central factor|Find other variations of central factor |

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

## Definition

### Symbol-free definition

A subgroup of a group is termed a **central factor of normalizer** if it satisfies the following equivalent conditinos:

- It is a central factor of its normalizer
- Every inner automorphism of the group that restricts to an automorphism on the subgroup restricts to an inner automorphism on the subgroup.

### Definition with symbols

A subgroup of a group is termed a **central factor of normalizer** if it satisfies the following equivalent conditions:

- where denotes the centralizer of in .
- For any in such that , there exists an element in such that for any in , .

## Formalisms

### First-order description

This subgroup property is a first-order subgroup property, viz., it has a first-order description in the theory of groups.

View a complete list of first-order subgroup properties

A subgroup of a group is a WC-subgroup if it satisfies the following condition:

### In terms of the in-normalizer operator

This property is obtained by applying the in-normalizer operator to the property: central factor

View other properties obtained by applying the in-normalizer operator

A subgroup is a WC-subgroup if and only if it is a central factor of its normalizer. This can be expressed in terms of thein-normalizer operator. The in-normalizer operator takes a subgroup property and outputs the property of being a subgroup that satisfies property inside the normalizer.

### In terms of the when-defined restriction formalism

The property of being a WC subgroup is the balanced property with respect to the when-defined function restriction formalism. Note that balanced properties for *when-defined* formalisms are not necessarily transitive.

## Relation with other properties

### Stronger properties

- Central factor: Its normalizer is the whole group, and it is clearly a central factor in that
- Self-normalizing subgroup: Its normalizer is itself, and it is clearly a central factor of itself.
- Cocentral subgroup

### Weaker properties

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

Check out the property operators part of this page for more details.

### Intersection-closedness

It seems unlikely that an intersection of WC-subgroups should again be a WC-subgroup.

### 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 and be a WC-subgroup of . Then, any inner automorphism of arises from an inner automorphism of , hence the restriction of an inner automorphism of to is also the restriction of an inner automorphism of to . Hence, it must be an inner automorphism of .

## Effect of property operators

### Right transiter

Any WC-subgroup of a central factor is a WC-subgroup. Thus, the right transiter of the property of being a WC-subgroup is weaker than the property of being a central factor. It's not clear whether it is exactly equal.

### Subordination

The property of being a WC subgroup is not transitive. Thus, there is a notion of sub-WC-subgroup.