On the Normal Subgroup with Exactly Two G-Conjugacy Class Sizes

Let G be a finite group with a non-central Sylow r-subgroup R, Z(G) the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either 1 or m. Particularly, it is shown that N is Abelian if N ∩ Z(G) = 1 and the G-conjugacy class size of every element of N is either 1 or m. Project supported by the National Natural Science Foundation of China (No. 10771132), SGRC (No. GZ 310), the Research Grant of Shanghai University and the Shanghai Leading Academic Discipline Project (No. J50101).