Central factor of normalizer

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

$H C_{G} (H) = N_{G} (H)$ where $C_{G} (H)$ denotes the centralizer of $H$ in $G$ .
For any $g$ in $G$ such that $g H g^{- 1} = H$ , there exists an element $h$ in $H$ such that for any $x$ in $G$ , $g x g^{- 1} = h x h^{- 1}$ .

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:

$\forall g \in G, (\exists h \in H . \forall x \in H, h x h^{- 1} = g x g^{- 1}) \lor (\exists x \in H . g x g^{- 1} \notin H)$

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.

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

Sub-WC-subgroup

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 $G \leq H \leq K$ 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.