Abstract
Let \( G \) be a finite group. A subgroup \( H \) of \( G \) is \( \mathfrak{U} \)-normal in \( G \) if every chief factor of \( G \) between \( H_{G} \) and \( H^{G} \) is cyclic; \( H \) is Sylow permutable in \( G \) if \( H \) commutes with every Sylow subgroup \( P \) of \( G \), i.e., \( HP=PH \). We say that a subgroup \( H \) of \( G \) is \( \mathfrak{U}\wedge sp \)-embedded in \( G \) if \( H=A\cap B \) for some \( \mathfrak{U} \)-normal subgroup \( A \) and Sylow permutable subgroup \( B \) in \( G \). We find the systems of subgroups \( \mathcal{L} \) in \( G \) such that \( G \) is supersoluble provided that each \( H\in\mathcal{L} \) is \( \mathfrak{U}\wedge sp \)-embedded in \( G \). In particular, we give new characterizations of finite supersoluble groups.
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Acknowledgment
The authors express their gratitude to the referee for useful remarks and suggestions.
Funding
The research was supported by the NNSF of China (nos. 12171126 and 12101165) and the Key Laboratory of Engineering Modeling and Statistical Computing of the Hainan Province. The work of the third author was carried out as part of the task of the State Program for Scientific Research “Convergence–2025” with financial support from the Ministry of Education of the Republic of Belarus (Project 20211328). The fourth author was supported by the Belarusian Republican Foundation for Basic Research (Grant no. F20R–291).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 3, pp. 626–638. https://doi.org/10.33048/smzh.2022.63.311
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Liu, AM., Guo, W., Safonova, I.N. et al. Lattice Characterizations of Finite Supersoluble Groups. Sib Math J 63, 520–529 (2022). https://doi.org/10.1134/S0037446622030119
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DOI: https://doi.org/10.1134/S0037446622030119
Keywords
- finite group
- Sylow permutable subgroup
- \( \mathfrak{U} \)-normal subgroup
- \( \mathfrak{U}\wedge sp \)-embedded subgroup
- supersoluble group