The influence of SΦ−supplemented subgroups on the structure of finite groups

Abstract

A subgroup H of a finite group G is said to be $S\Phi -$supplemented in G if there exists a subnormal subgroup T of G such that G = HT and $H\cap T\le \Phi (H),$ where $\Phi (H)$ is the Frattini subgroup of H. In this article, we investigate the structure of a finite group G under the assumption that certain subgroups of G are $S\Phi -$supplemented in G. We obtain that a finite group G is nilpotent if and only if every Sylow subgroup of G is $S\Phi -$supplemented in G. And a group G is nilpotent if every maximal subgroup of G is $S\Phi -$supplemented in G. Moreover, the commutativity of G is characterized by using that the minimal subgroups of two non–conjugate maximal subgroups of G are $S\Phi -$supplemented in G.

Keywords:

• S$\Phi -$supplemented subgroup
• abelian group
• nilpotent group
• supersolvable group

2020 Mathematics Subject Classification:

• Primary: 20D10
• 20D15
• 20D20

Acknowledgements

The authors are very grateful to the referee for her/his valuable suggestions and useful comments. They would like to thank the referee for providing simpler proofs of Theorems 3.4, 3.5 and 3.14.

Additional information

Funding

This research was supported by a grant of the Natural Science Foundation of Guangxi Province (No. 2021GXNSFAA220105, No. 2020GXNSFDA238014).