On Φ-Π-property of subgroups of finite groups

Abstract

Let H be a subgroup of a finite group G. We say that H satisfies $\Phi$-$\Pi$-property in G if for any non-Frattini chief factor L/K of G, $|G/K:N_{G/K}(HK/K\cap L/K)|$ is a $\pi (HK/K\cap L/K)$-number. In this paper, we study the influence of some subgroups of G satisfying $\Phi$-$\Pi$-property on the structure of G, and obtain some interesting results.

Keywords:

• Finite group
• formation
• $\Phi$-$\Pi$-property
• p-supersoluble group

2020 Mathematics Subject Classification:

• 20D10
• 20D20

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (Grant No.12071376, 11971391), the Fundamental Research Funds for the Central Universities (No. XDJK2020B052), the Natural Science Foundation Project of CQ (No.cstc2021jcyj-msxmX0426) and the Fundamental Research Funds for the Central Universities (Nos.XDJK2019C116 and XDJK2019B030).