Equivalence Classes with a Single Element

Hi guys, question here. It's not for a class.

"Let be a group. Then the relation if and only if for some is an equivalence relation on . Some equivalence classes contain only one element . Characterize those elements ."

It's pretty clear that if is in the center of , then must be in a class by itself since means so that .

But could there be more? This question comes after a discussion of the commutator, and the book hasn't even discussed center yet, so I'm trying to figure out what kind of answer the author was after.