C-closed subgroup

Symbol-free definition
A subgroup of a group is termed a c-closed subgroup if it satisfies the equivalent conditions:


 * It equals the centralizer of its centralizer.
 * It occurs as the centralizer of some subset of the group
 * It occurs as the centralizer of some subgroup

Alternative terminology for a c-closed subgroup is centralizer subgroup, self-bicommutant subgroup, self-bicentralizer subgroup.

Formalisms
The property of being a centralizer subgroup is a, with respect to the Galois correspondence induced by relation of commuting.

Metaproperties
A c-closed subgroup of a c-closed subgroup is again c-closed.

An arbitrary intersection of centralizer subgroups is a centralizer subgroup; this follows from general facts about Galois correspondences. In fact, even an empty intersection of centralizer subgroups is a centralizer subgroups, so the property is actually strongly intersection-closed.