# 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:

### Definition with symbols

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

• $HC_G(H) = N_G(H)$ where $C_G(H)$ denotes the centralizer of $H$ in $G$.
• For any $g$ in $G$ such that $gHg^{-1} = H$, there exists an element $h$ in $H$ such that for any $x$ in $G$, $gxg^{-1} = hxh^{-1}$.

## 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 $H$ of a group $G$ is a WC-subgroup if it satisfies the following condition: $\forall g \in G, (\exists h \in H . \forall x \in H, hxh^{-1} = gxg^{-1}) \lor (\exists x \in H . gxg^{-1} \notin H)$

### 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 $p$ and outputs the property of being a subgroup that satisfies property $p$ 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.

## Metaproperties

### Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the 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 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 condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $G \le H \le K$ and $G$ be a WC-subgroup of $K$. Then, any inner automorphism of $H$ arises from an inner automorphism of $K$, hence the restriction of an inner automorphism of $H$ to $G$ is also the restriction of an inner automorphism of $K$ to $G$. Hence, it must be an inner automorphism of $G$.

## 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.