C-closed subgroup

In mathematics, in the field of group theory a subgroup of a group is said to be c-closed if any two elements of the subgroup that are conjugate in the group are also conjugate in the subgroup.

An alternative characterization of c-closed normal subgroups is that all class automorphisms of the whole group restrict to class automorphisms of the subgroup.

The following facts are true regarding c-closed subgroups:

• Every central factor (a subgroup that may occur as a factor in some central product) is a c-closed subgroup.
• Every c-closed normal subgroup is a transitively normal subgroup.
• The property of being c-closed is transitive, that is, every c-closed subgroup of a c-closed subgroup is c-closed.

The property of being c-closed is sometimes also termed as being conjugacy stable. It is a known result that for finite field extensions, the general linear group of the base field is a c-closed subgroup of the general linear group over the extension field. This result is typically referred to as a stability theorem.

A subgroup is said to be strongly c-closed if all intermediate subgroups are also c-closed.
External links

C-closed subgroup at the Group Properties Wiki
Central factor at the Group Properties Wiki

Mathematics Encyclopedia