MSC: 20D10; 20E28
DOI: 10.21538/0134-4889-2023-29-1-254-258
This work was supported by the Ministry of Education of the Republic of Belarus (Grant number 20211467).
Полный текст статьи (Full text)
A subgroup $M$ of a group $G$ is an $n$-maximal subgroups of $G$ if there is a subgroup chain $M=M_n\leq M_{n-1}\leq \ldots \leq M_1\leq M_0=G$ such that $M_{i+1}$ is a maximal subgroup of $M_i$. We establish a criterion for a group with absolutely $\mathfrak{F}$-subnormal $n$-maximal subgroups to belong to a subgroup-closed saturated formation $\mathfrak{F}$ containing all nilpotent groups.
Keywords: finite group, maximal subgroup, subnormal subgroup
REFERENCES
1. Huppert B. Normalteiler and maximal Untergruppen endlicher Gruppen. Math. Z., 1954, vol. 60, pp. 409–434. doi: 10.1007/BF01187387
2. Janko Z. Finite groups with invariant fourth maximal subgroups. Math. Z., 1963, vol. 82, pp. 82–89. doi: 10.1007/BF01112825
3. Mann A. Finite groups whose n-maximal subgroups are subnormal. Trans. Amer. Math. Soc., 1968, vol. 132, pp. 395–409.
4. Vasil’ev A. F., Melchenko A. G. Finite groups with absolutely formationally subnormal Sylow subgroups. Probl. Fiz. Math. Tekh., 2019, vol. 4, no. 41, pp. 44–50 (in Russian).
5. Konovalova M.N., Monakhov V.S., Sokhor I.L. Finite groups with formational subnormal strictly 2-maximal subgroups. Comm. Algebra, 2022, vol. 50, no. 4, pp. 1606–1612. doi: 10.1080/00927872.2021.1986058
6. Kovaleva V.A., Skiba A.N. Finite solvable groups with all n-maximal subgroups $\mathfrak{F}$-subnormal. J. Group Theory, 2014, vol. 17, no. 3, pp. 273–290. doi: 10.1515/jgt-2013-0047
7. Kovaleva V.A., Yi X. Finite biprimary groups with all 3-maximal subgroups $\mathfrak{U}$-subnormal. Acta Math. Hung., 2015, vol. 146, no. 1, pp. 47–55. doi: 10.1007/s10474-015-0498-5
8. Huppert B. Endliche Gruppen I. Berlin: Springer-Verl., 1967. 793 p. doi: 10.1007/978-3-642-64981-3
9. Burness T.C., Liebeck M.W., Shalev A. On the length and depth of finite groups. Proc. London Math. Soc., 2019, vol. 119, no. 3, pp. 1464–1492. doi: 10.1112/plms.12273
10. Sokhor I.L. Continuation of the theory of $E_{\mathfrak{F}}$-groups.Trudy Inst. Mat. i Mekh. UrO RAN, 2021, vol. 27, no. 1, pp. 268 -272.doi : 10.21538∕0134 - 4889 - 2021 - 27 - 1 - 268 - 272
11. Kohler J. A note on solvable groups. J. Lond. Math. Soc., 1968, vol. 43, pp. 235–236
12. Iwasawa K. Über die endlichen Gruppen und die Verbände ihrer Untergruppen. J. Fac. Sci. Imp. Univ. Tokyo. Sect. I., 1941, vol. 4, pp. 171–199.
13. Monakhov V.S. Schmidt subgroups, their existence and some applications. Proceedings of Ukrainian Mathematical Congress–2001. Inst. Mat. NAN Ukrainy, Kyiv, 2002, pp. 81–90 (in Russian).
Received November 9, 2022
Revised January 20, 2023
Accepted January 30, 2023
Funding Agency: This work was supported by the Ministry of Education of the Republic of Belarus (Grant number 20211467).
Irina Leonidovna Sokhor, Can. Sci. (Phys.-Math.), Francisk Skorina Gomel State University, 246019 Gomel, Belarus, e-mail: irina.sokhor@gmail.com
Cite this article as: I.L. Sokhor. Finite groups with absolutely $\mathfrak{F}$-subnormal maximal subgroups, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 254–258.
Русский
И.Л. Сохор. Конечные группы с абсолютно $\mathfrak{F}$-субнормальными максимальными подгруппами
Подгруппа $M$ группы $G$ является $n$-максимальной подгруппой в группе $G$, если существует цепочка подгрупп $M=M_n\leq M_{n-1}\leq \ldots \leq M_1\leq M_0=G$ такая, что $M_{i+1}$ — максимальная подгруппа в $M_i$. Для группы с абсолютно $\mathfrak{F}$-субнормальными $n$-максимальными подгруппами установлен критерий принадлежности наследственной насыщенной формации $\mathfrak{F}$, содержащей все нильпотентные группы.
Ключевые слова: конечная группа, максимальная подгруппа, субнормальная подгруппа
