On an open problem of Guo-Skiba

摘要

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 (A a (c) H)/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.