Abstract
Granted a partition \( \sigma \) of the set of all primes, we study the structure of a finite group with a given system of \( \sigma \)-subnormal Schmidt subgroups.
This is a preview of subscription content, log in via an institution to check access.
References
The Kourovka Notebook: Unsolved Problems in Group Theory. 19th ed., Khukhro E.I. and Mazurov V.D. (eds.), Sobolev Inst. Math., Novosibirsk (2018).
Guo W., Safonova I.N., and Skiba A.N., “On \( \sigma \)-subnormal subgroups of finite groups,” Southeast Asian Bull. Math., vol. 45, no. 6, 813–824 (2021).
Skiba A.N., “On \( \sigma \)-subnormal and \( \sigma \)-permutable subgroups of finite groups,” J. Algebra, vol. 436, 1–16 (2015).
Arad Z. and Chillag D., “A criterion for the existence of normal \( \pi \)-complements in finite groups,” J. Algebra, vol. 87, 472–482 (1984).
Vedernikov V.A., “On \( \pi \)-properties of finite groups,” in: Arithmetic and Subgroup Structure of Finite Groups, Nauka i Tekhnika, Minsk (1986), 13–19 [Russian].
Knyagina V.N. and Monakhov V.S., “Finite groups with subnormal Schmidt subgroups,” Sib. Math. J., vol. 45, no. 6, 1075–1079 (2004).
Vedernikov V.A., “Finite groups with subnormal Schmidt subgroups,” Algebra Logic, vol. 46, no. 6, 363–372 (2007).
Al-Sharo K.A. and Skiba A.N., “On finite groups with \( \sigma \)-subnormal Schmidt subgroups,” Comm. Algebra, vol. 45, no. 10, 3117–3134 (2017).
Hu B. and Huang J., “On finite groups with generalized \( \sigma \)-subnormal Schmidt subgroups,” Comm. Algebra, vol. 46, no. 7, 3127–3134 (2018).
Sun F., Yi X., and Kamornikov S.F., “Finite groups with generalized subnormal Schmidt subgroups,” Sib. Math. J., vol. 62, no. 2, 364–369 (2021).
Yi X. and Kamornikov S.F., “Finite groups with \( \sigma \)-subnormal Schmidt subgroups,” J. Algebra, vol. 560, 181–191 (2020).
Ballester-Bolinches A., Kamornikov S.F., and Yi X., “On \( \sigma \)-subnormality criteria in finite groups,” J. Pure Appl. Algebra, vol. 226, no. 2, Article 106822 (2022).
Doerk K. and Hawkes T., Finite Soluble Groups, De Gruyter, Berlin and New York (1992).
Schmidt O. Yu., “Groups whose all subgroups are special,” Mat. Sb., vol. 31, no. 3, 366–372 (1924).
Golfand Yu.A., “On groups all subgroups of which are special,” Dokl. Akad. Nauk SSSR, vol. 60, no. 8, 1313–1315 (1948).
Hu B., Huang J., and Skiba A.N., “Characterizations of finite \( \sigma \)-nilpotent and \( \sigma \)-quasinilpotent groups,” Bull. Malays. Math. Soc., vol. 42, no. 5, 2091–2104 (2019).
Kamornikov S.F. and Selkin M.V., Subgroup Functors and Classes of Finite Groups, Belarusskaya Nauka, Minsk (2003) [Russian].
Kazarin L.S., Martinez-Pastor A., and Pérez-Ramos M.D., “On the Sylow graph of a group and Sylow normalizers,” Isr. J. Math., vol. 186, 251–271 (2011).
Shemetkov L.A., Formations of Finite Groups, Nauka, Moscow (1978) [Russian].
Funding
The first author was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY18A010028). The third author was supported by the Ministry of Education of the Republic of Belarus (Grant no. 20211779 “Convergence–2025”).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 1, pp. 89–97. https://doi.org/10.33048/smzh.2023.64.109
To the blessed memory of Viktor Aleksandrovich Vedernikov.
Rights and permissions
About this article
Cite this article
Yi, X., Li, M. & Kamornikov, S.F. Finite Groups with a System of Generalized Subnormal Schmidt Subgroups. Sib Math J 64, 76–82 (2023). https://doi.org/10.1134/S0037446623010093
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446623010093
Keywords
- finite group
- \( \sigma \)-subnormal subgroup
- Schmidt group
- \( \sigma \)-nilpotent group
- \( \sigma \)-hypercenter