C-closed normal subgroup

Symbol-free definition
A subgroup of a group is termed a c-closed normal subgroup if it is both a c-closed subgroup (it equals the centralizer of its centralizer) and a normal subgroup (it is invariant under inner automorphisms).

Stronger properties

 * Weaker than::c-closed central factor
 * Weaker than::c-closed characteristic subgroup
 * Weaker than::c-closed transitively normal subgroup

Metaproperties
An arbitrary intersection of c-closed normal subgroups is c-closed normal. This follows from the corresponding facts being true for c-closed subgroups and normal subgroups individually.

The centralizer of a c-closed normal subgroup is again c-closed normal. This again follows from the corresponding statements for both properties.