On m-S-supplemented subgroups of finite groups

Abstract

Let G be a finite group and H a subgroup of G. We say that H is m-S-permutable in G, if $H=〈A,B〉$ for some modular subgroup A and S-permutable subgroup B of G; m-S-supplemented in G if there are an m-S-permutable subgroup S and a subgroup T of G such that G = HT and $H\cap T\le S\le H$. In this article, we study finite groups with given systems of m-S-supplemented subgroups. In particular, we prove that if E is a normal subgroup of G such that for any noncyclic Sylow subgroup P of E all maximal subgroups of P or all cyclic subgroups of P of prime order and order 4 (in the case when P is a non-Abelian 2-group) are m-S-supplemented in G, then E is hypercyclically embedded in G.

Keywords:

• Finite group
• modular subgroup
• m-S-permutable subgroup
• m-S-supplemented subgroup
• S-permutable subgroup

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D15
• 20D30

Acknowledgment

The authors are very grateful to the helpful suggestions of the referee.

Additional information

Funding

Research is supported by a NNSF grant of China (Grant #11301227).