Central subgroup of normalizer

Definition
A subgroup of a group is termed a central subgroup of normalizer if it satisfies the following equivalent conditions:


 * It is a defining ingredient::central subgroup (i.e., is contained in the defining ingredient::center) of its normalizer.
 * It is Abelian and is a central factor of normalizer.
 * Any inner automorphism of the whole group that leaves the subgroup invariant, must act trivially on the subgroup.

Facts

 * Burnside's normal p-complement theorem: This states that if a Sylow subgroup is a central subgroup of its normalizer, then it is a retract, i.e., it possesses a normal complement.