The finite groups with IC-property of subgroups

Abstract

A subgroup H of a group G is called an $\mathit{\text{ICΦs}}$-subgroup of G if $(H\cap [H,G])H_G/H_G$ $\le \Phi (H/H_G)H_{sG}/H_G,$ where $H_G=\underset{g\in G}{\cap }{H}^g$ and $\Phi (H/H_G)$ is the Frattini subgroup of $H/H_G$ and H_sG is the largest s-permutable subgroup of G contained in H. In this paper, we investigate the structure of finite group G under the assumption that some subgroups of G are $\mathit{\text{ICΦs}}$-subgroups of G, some results are obtained.

Key words:

• finite group, Sylow subgroup, $\mathit{\text{ICΦs}}$-subgroup, p-nilpotent

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D10
• 20D20

Acknowledgement

The authors are grateful to the referee for valuable comments and detailed corrections.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China(Grant N. 11871360, 11901417)