On finite groups factorized by σ-nilpotent subgroups

Abstract

Let G be a finite group and $\sigma =\left\{{\sigma }_i\right|i\in I\}$ be a partition of the set of all primes $\mathbb{P},$ that is, $\mathbb{P}=\underset{i\in I}{\cup }{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j.$ A chief factor H/K of G is said to be σ-central in G if the semidirect product $(H/K)⋊(G/C_G(H/K))$ is a σ_i-group for some $i\in I.$ The group G is said to be σ-nilpotent if either G = 1 or every chief factor of G is σ-central in G. In this article, we study the properties and structures of a finite group G = AB, factorized by two σ-nilpotent subgroups A and B. Some known results are generalized.

Keywords:

• finite group
• nilpotent subgroup
• σ-central
• σ-nilpotent subgroup

2020 Mathematics Subject Classification:

• 20D10
• 20D15
• 20D20

Acknowledgments

The authors thank the referees for their careful reading and helpful comments.

Additional information

Funding

Research was supported by the Natural Science Foundation of China (No. 12171126,12001526), Fundamental Research Funds for the Central Universities (JUSRP121048), Natural Science Foundation of Jiangsu Province, China (No. BK20210442,BK20200626) and Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province.