Conjugacy-closed normal subgroup

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 $$H$$ of a group $$G$$ is termed conjugacy-closed normal if it satisfies the following equivalent conditions:


 * $$H \triangleleft G$$ and whenever $$gxg^{-1} = y$$ for $$x,y \in H$$ and $$g \in G$$, there exists $$h \in H$$ such that $$hxh^{-1} = y$$.


 * Given any automorphism $$\sigma$$ of $$G$$ such that for every $$x \in G$$, there exists $$g \in G$$ such that $$\sigma(x) = gxg^{-1}$$, we also have the following: for every $$x \in H$$ there exists $$h \in H$$ such that $$\sigma(x) = hxh^{-1}$$.

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Central factor
 * Weaker than::Central subgroup
 * Weaker than::Cocentral subgroup

Weaker properties

 * Stronger than::Conjugacy-closed subgroup
 * Stronger than::Normal subgroup
 * Stronger than::Transitively normal subgroup

Metaproperties
if $$H$$ is conjugacy-closed normal in $$K$$ and $$K$$ is conjugacy-closed normal in $$G$$, then $$H$$ is conjugacy-closed normal in $$G$$. 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.

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.

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


 * If $$H$$ is conjugacy-closed in $$G$$, $$H$$ is conjugacy-closed in any intermediate subgroup $$K$$ of $$G$$.
 * If $$H$$ is normal in $$G$$, $$H$$ is noraml in every intermediate subgroup $$K$$ of $$G$$.