On Small P-part Of Relative Character Degree Over A Normal Subgroup

Let Z be a normal subgroup of a finite group G,?be an irreducible character of Z.A recent result says if for any irreducible character?of G which lies over?,p(?)?(1)/?(1),then G/Z has abelian Sylow p-subgroups.In this article,we consider the case p~2(?)?(1)/?(1)for any irreducible character?of G which lies over?.We want to determine the structure of the Sylow p-subgroups of G/Z.For the case Z=1,G is a p-group,the structure of G is known,however for a general G,its Sylow p-subgroups may not have this structure(at least for some particular p).We prove a result for Sylow p-subgroups of G/Z in case Z=1,and similar results for the relative case under some conditions.This relies on some results of the case G/Z is simple,roughly speaking,in this case,Sylow p-subgroups of G/Z may have small order,and they are abelian for few exceptions.