On the orders of conjugacy classes of cyclic subgroups

Introduction. In recent years, there have been interests in studying the influence of arithmetic conditions on the lengths of conjugacy classes to the structure of finite groups. The results are parallel to those of character degrees, and lead to strong restrictions to the group structure. An interesting relevant topic is the lengths of conjugacy classes of subgroups. This was first brought up to the author by Professor J. P. Zhang, who has obtained some interesting results on conjugacy class lengths of Sylow subgroups, the so called Sylow numbers [91. In this paper, we consider conjugacy class lengths of cyclic subgroups of finite goups. Our notations are standard and tends to follow [5]. All groups considered are finite. Let G be a finite group, H a subgroup of G, then the length of conjugacy class of H, the number of subgroups of G conjugate to H, is equal to I G : NG(H)[. Our main result is Theorem 1, its proof depends on the classification of finite simple groups.