On an open problem of Guo-Skiba

Abstract

Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/AG of G is called a G-boundary factor of A. For any Gboundary factor H/AG of A, the subgroup (A ∩ H)/AG of G/AG 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 pʹ-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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Higher Education Press and Springer-Verlag Berlin Heidelberg