Abstract
Let \(\pi\) be a set of primes. A subgroup \(H\) of a group \(G\) is said to be \(\mathbb{P}_{\pi}\)-subnormal in \(G\) if either \(H=G\) or there exists a chain of subgroups beginning with \(H\) and ending with \(G\) such that the index of each subgroup in the chain is either a prime in \(\pi\) or a \(\pi'\)-number. Properties of \(\mathbb{P}_{\pi}\)-subnormal subgroups are studied. In particular, it is proved that the class of all \(\pi\)-closed groups in which all Sylow subgroups are \(\mathbb{P}_{\pi}\)-subnormal is a hereditary saturated formation. Criteria for the \(\pi\)-supersolvability of a \(\pi\)-closed group with given systems of \(\mathbb{P}_{\pi}\)-subnormal subgroups are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. S. Kazarin, “On groups with factorization,” Dokl. Akad. Nauk SSSR 256 (1), 26–29 (1981).
A. F. Vasil’ev, T. I. Vasil’eva, and V. N. Tyutyanov, “On the finite groups of supersoluble type,” Siberian Math. J. 51 (6), 1004–1012 (2010).
A. F. Vasil’ev, T. I. Vasil’eva, and V. N. Tyutyanov, “On the products of \(\mathbb P\)-subnormal subgroups of finite groups,” Siberian Math. J. 53 (1), 47–54 (2012).
V. N. Kniahina and V. S. Monakhov, “On supersolvability of finite groups with \(\mathbb{P}\)-subnormal subgroups,” Int. J. of Group Theory 2 (4), 21–29 (2013).
V. I. Murashka, “Classes of finite groups with generalized subnormal cyclic primary subgroups,” Siberian Math. J. 55 (6), 1105–1115 (2014).
V. A. Vasilyev, “Finite groups with submodular Sylow subgroups,” Siberian Math. J. 56 (6), 1019–1027 (2015).
A. Ballester-Bolinches, Y. Li, M. C. Pedraza-Aguilera, and N. Su, “On factorised finite groups,” Mediterr. J. Math. 17 (2), 65 (2020).
B. Huppert, Endliche Gruppen. I (Springer- Verlag, Berlin, 1967).
L. A. Shemetkov, Formations of Finite Groups (Nauka, Moscow, 1978) [in Russian].
M. Bianchi, A. Gillio Berta Mauri, and P. Hauck, “On finite groups with nilpotent Sylow-normalizers,” Arch. Math. (Basel) 47 (3), 193–197 (1986).
T. I. Vasil’eva and A. G. Koranchuk, “Finite groups with subnormal residuals of Sylow normalizers,” Siberian Math. J. 63 (4), 670–676 (2022).
K. Doerk and T. Hawkes, Finite Soluble Groups (Walter de Gruyter, Berlin–New York, 1992).
A. F. Vasil’ev, “New properties of finite dinilpotent groups,” Vestsi Nats. Akad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk, No. 2, 39–43 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 483–496 https://doi.org/10.4213/mzm13973.
Rights and permissions
About this article
Cite this article
Vasil’eva, T.I., Koranchuk, A.G. On Finite Groups with \(\mathbb{P}_{\pi}\)-Subnormal Subgroups. Math Notes 114, 421–432 (2023). https://doi.org/10.1134/S0001434623090158
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434623090158