The groups which have exactly one conjugacy class of centralizers of elements are well known. They are the abelian groups.

More generally, in this paper, we will study groups with finitely many conjugacy classes of centralizers and we will investigate properties of groups with two and three conjugacy classes of centralizers. Problems like these were studied, in the finite case, by many authors ([J. London Math. Soc. (2) 6 (1973), 421-426], [Nagoya Math. J. 53 (1974), 47-57], [N. It?n Finite Groups with given Conjugate Types I, Nagoya University, 1952, pp. 17-28], [Osaka J. Math. 7 (1970), 231-251]). Here, we will extend their study to the infinite case and we will get some further information in the finite case.

centralizers, conjugacy classes, Frobenius groups, central-by-finite groups.