Central factor of normalizer

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:


 * $$HC_G(H) = N_G(H)$$ 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
A subgroup $$H$$ of a group $$G$$ is a WC-subgroup if it satisfies the following condition:

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

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.

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
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.

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 $$G$$.

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.