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On an open problem of Guo-Skiba
Authors:Zhenfeng Wu  Wenbin Guo  Baojun Li
Affiliation:1. Department of Mathematics, University of Science and Technology of China, Hefei 230026, China2. College of Applied Mathematics, Chengdu University of information Technology, Chengdu 610225, China
Abstract:Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/A G of G is called a G-boundary factor of A. For any Gboundary factor H/A G of A, the subgroup (AH)/A G of G/ A G is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups.
Keywords:Finite group  p-hypercyclically embedded subgroup  G-boundary factor  G-trace of subgroup  meet-irreducible subgroup  
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