Primary subgroups and the structure of finite groups

Abstract

Let G be a group and $H\le G.$ The permutizer of H in G is $P_G(H)=〈x|x\in G,〈x〉H=H〈x〉〉.$ H is said to be strongly permuteral in G if $P_U(H)=U$ whenever $H\le U\le G.$ Moreover, let $P_{G_P}(H)=〈x|x$ is a primary element of $G,〈x〉H=H〈x〉〉.$ H is said to be strongly $\mathbb{P}$-permuteral in G if $P_{U_P}(H)=U$ whenever $H\le U\le G.$ In this article, we study the structure of a group G in which every Sylow subgroup and its maximal subgroups are strongly $\mathbb{P}$-permuteral or abnormal and the structure of G with self-normalizing or strongly permuteral Sylow subgroups.

Keywords:

• Abnormal subgroup
• self-normalizing subgroup
• strongly permuteral
• strongly $\mathbb{P}$-permuteral

2010 AMS Subject Classification:

• 20D10
• 20D20

Acknowledgments

The authors are indebted to the referees for valuable suggestions and very careful reading of the original manuscript.

Additional information

Funding

The research of the work is supported by the National Natural Science Foundation of China (11901169), the Youth Science Foundation of Henan Normal University (2019QK02) and the project for high quality courses of postgraduate education in Henan Province, research and practice project of higher education reform in Henan Normal University (post-graduate education, No. YJS2019JG06).