On nearly M-supplemented primary subgroups of finite groups

Abstract

A subgroup H is called to be nearly $\mathcal{M}$-supplemented in G if G has a normal subgroup K such that HK ⊴ G and $H_{i}K<HK$ for every maximal subgroup H_i of H. The main goal of this paper is to investigate the structure of chief factors of finite groups by using nearly $\mathcal{M}$-supplemented primary subgroups and obtain some new characterization about chief factors of finite groups. The main result is the following: Let $G\in E_{p\prime }$ and P be a Sylow p-subgroup of G, where p is an odd prime. If every maximal subgroup of P is nearly $\mathcal{M}$-supplemented in G, then every non-abelian pd-G-chief factor A/B satisfies one of the following conditions:

1. $A/B\cong PSL(2,7)$ and p = 7; $A/B\cong PSL(2,11)$ and p = 11;

2. $A/B\cong PSL(2,2^t)$ and $p=2^t+1>3$ is a Fermat prime;

3. $A/B\cong PSL(n,q),n\ge 3$ is a prime, $(n,q-1)=1$ and $p=q^n-1/q-1$;

4. $A/B\cong M_{11}$ and p = 11; $A/B\cong M_{23}$ and p = 23;

5. $A/B\cong A_p$ and $p\ge 5$.

Keywords:

• maximal subgroup
• nearly $\mathcal{M}$-supplemented subgroup
• simple group
• chief factor

2010 Mathematics Subject Classification:

• 20D10
• 20D20

Acknowledgements

We thank the reviewers for their suggestions which have helped to improve our original version.

Additional information

Funding

This research is supported by the grant of NSFC (Grant # 11271016 and 11501235) and the Key Natural Science Foundation of Anhui Education Commission (KJ2017A569) and the Fundamental Research Funds of China West Normal University (17E091).