Let a group acting on itself by conjugation. Let be a subset of . The stabilizer of is called the centralizer of and it's the set

   for all

For any group , , the center of . Thus, any subgroup of is an abelian subgroup of . However, the converse is generally not true. For example, take any non-abelian group and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.