p-divisibility of conjugacy class sizes and normal p-complements

Let N be a normal subgroup of a group G and let p be a prime.We prove that if the p-part of jxGj is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement.This research is supported by the Valencian Government, Proyecto PROMETEO/2011/30, and by the Spanish Government, Proyecto MTM2010-19938-C03-02. The third author is supported by the research Project NNSF of China (Grant No. 11201401 and No. 11301218) and University of Jinan Research Funds for Doctors (XBS1335 and XBS1336)