C closed subgroup

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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 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.