A note on maximal subgroups of σ-soluble groups

Abstract

Let $\sigma =\{{\sigma }_i:i\in I\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A finite group G is called a σ-soluble group if every chief factor of G is a σ_i-group for some ${\sigma }_i\in \sigma$. In this note, we investigate how maximal subgroups of a σ-soluble group G are embedded in G. Some criteria for σ-solubility are obtained, which generalize some classical results.

Keywords:

• Finite group
• Finite σ-soluble group
• σ-c-Normal subgroup

2020 Mathematics Subject Classification:

• 20D10
• 20D20

Acknowledgments

The authors are very grateful to the referees for the helpful comments and suggestions.

Additional information

Funding

This work is supported in part by NSF of China (Grants No. 12071093, 12026212, 12071092, 12101339) and NSF of Guangdong Province (Grant No. 2021A1515010217).