有限群共轭类长的一个问题

On a Problem of the Length of Conjugacy Classes of Finite Groups

设A和B都是有限群G的子群且G=AB.若A是G的次正规子群,且对每个p∈π(G)以及每个素数幂阶的p′-元x∈A∪B,p^2均不整除|x^G|,则G为超可解群.这个结果正面解答了由石向东,韦华全和马儇龙于2013年提出的一个问题,统一推广了由刘晓蕾于2011年得到的三个定理.

Let G be a finite group with subgroups A and B such that G = AB. If A is subnormal in G and for any p ∈π （G） and any p＇-element x of A ∪B with prime power order, ｜xG｜ is not divisible by p2, then G is supersolvable. This result gives a positive answer to a problem posed in 2013 by Shi Xiangdong, Wei Huaquan and Ma Xuanlong and a unify generalization of three theorems obtained in 2011 by Liu Xiaolei.