$ s^{\ast} $-normality and coprime action in finite groups

Xiaoxia Dong; Yubo Lv; Hui Wu; Xiaoyan Sheng; Xiaoxia Dong; Yubo Lv; Hui Wu; Xiaoyan Sheng

doi:10.3934/era.2026253

Electronic Research Archive

2026, Volume 34, Issue 8: 5689-5702. doi: 10.3934/era.2026253

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Research article

$ s^{\ast} $-normality and coprime action in finite groups

1.
Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China
2.
School of Mathematics, Guangxi University, Nanning 530004, China

Received: 02 May 2026 Revised: 22 June 2026 Accepted: 25 June 2026 Published: 06 July 2026

A subgroup $ H $ of a finite group $ G $ is called $ s^{\ast} $-normal in $ G $ if some normal subgroup $ T $ of $ G $ satisfies $ H_G\le T $, $ G=HT $, and the quotient $ (H\cap T)/H_G $ has square-free order, where $ H_G $ is the core of $ H $ in $ G $. This paper gave several new criteria for finite groups to be solvable by considering the $ s^{\ast} $-normality of certain $ n $-maximal invariant subgroups under coprime actions, where $ n\in\{1,2,3\} $.
- solvable group,
- n-maximal subgroup,
- $ s^{\ast} $-normal subgroup,
- coprime action
Citation: Xiaoxia Dong, Yubo Lv, Hui Wu, Xiaoyan Sheng. $ s^{\ast} $-normality and coprime action in finite groups[J]. Electronic Research Archive, 2026, 34(8): 5689-5702. doi: 10.3934/era.2026253

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Abstract

A subgroup $ H $ of a finite group $ G $ is called $ s^{\ast} $-normal in $ G $ if some normal subgroup $ T $ of $ G $ satisfies $ H_G\le T $, $ G=HT $, and the quotient $ (H\cap T)/H_G $ has square-free order, where $ H_G $ is the core of $ H $ in $ G $. This paper gave several new criteria for finite groups to be solvable by considering the $ s^{\ast} $-normality of certain $ n $-maximal invariant subgroups under coprime actions, where $ n\in\{1,2,3\} $.

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