In mathematical group theory, a subgroup of a group is termed a special abelian subgroup or SA-subgroup if the centralizer of any nonidentity element in the subgroup is precisely the subgroup(Curtis & Reiner 1981, p.354). Equivalently, an SA subgroup is a centrally closed abelian subgroup.
SA subgroups are known for certain characters associated with them termed exceptional characters.
References
Special abelian subgroup Wikipedia(Text) CC BY-SA