On Finite Groups with Special Conjugacy Classes

摘要

Let G be a finite group with the property that for any conjugacy class order, G has exactly two conjugacy classes which have the same order. We prove that; (1) if a Sylow 2-subgroup of G is Abelian, then G is isomorphic to the direct product of symmetric group with order 3 and cyclic group with order 2, or G is isomorphic to the semidirect product of a cyclic group with order 3 and a cyclic group with order4; (2) if G' is nilpotent, then G is a group of {2,3,5}.