Geodesic
On $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups of finite groups
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 4, pp. 526-535 Cet article a éte moissonné depuis la source Math-Net.Ru

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Only finite groups are considered. $\frak F$-projectors and $\frak F$-covering subgroups, where $\frak F$ is a certain class of groups, were introduced into consideration by W. Gaschutz as a natural generalization of Sylow and Hall subgroups in finite groups. Developing Gaschutz's idea, V. A. Vedernikov and M. M. Sorokina defined $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, where $\omega$ is a non-empty set of primes, and established their main characteristics. The purpose of this work is to study the properties of $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups, establishing their relation with other subgroups in groups. The following tasks are solved: for a non-empty $\omega$-primitively closed homomorph $\frak F$ and a given set $\pi$ of primes, the conditions under which an $\frak F^{\omega}$-projector of a group coincides with its $\pi$-Hall subgroup are established; for a given formation $\frak F$, a relation between $\frak F^{\omega}$-covering subgroups of a group $G=A\rtimes B$ and $\frak F^{\omega}$-covering subgroups of the group $B$ is obtained. In the paper classical methods of the theory of finite groups, as well as methods of the theory of classes of groups are used. 
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M. M. Sorokina; D. G. Novikova. On $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups of finite groups. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 24 (2024) no. 4, pp. 526-535. http://geodesic.mathdoc.fr/item/ISU_2024_24_4_a4/

[1] Gaschütz W., “Zur theorie der endlichen auflösbaren Gruppen”, Mathematische Zeitschrift, 80:4 (1962), 300–305 (in German) | DOI

[2] Schunck H., “$\frak H$-Untergruppen in endlichen auflösbaren gruppen”, Mathematische Zeitschrift, 97:4 (1967), 326–330 (in German) | DOI

[3] Shemetkov L. A., “On the product of formations”, Doklady AN BSSR, 28:2 (1984), 101–103 (in Russian)

[4] Vedernikov V. A., Sorokina M. M., “$\frak F$-projectors and $\frak F$-covering subgroups of finite groups”, Siberian Mathematical Journal, 57:6 (2016), 957–968 | DOI | DOI

[5] Gaschütz W., Lectures on Subgroups of Sylow type in finite soluble groups, Notes on pure mathematics, 11, Australian National University, Canberra, 1979, 100 pp.

[6] Shemetkov L. A., Formations of finite groups, Nauka, M., 1978, 272 pp. (in Russian)

[7] Doerk K., Nawkes T., Finite soluble groups, Walter de Gruyter, Berlin–New York, 1992, 891 pp.

[8] Vasil'eva T. I., Prokopenko A. I., “Projectors and lattices of normal subgroups of finite groups”, Doklady of the National Academy of Sciences of Belarus, 48:4 (2004), 34–37 (in Russian)

[9] Erickson R., “Projectors of finite groups”, Communication in Algebra, 10:18 (1982), 1919–1938 | DOI

[10] Förster P., “Projektive Klassen endlicher Gruppen I. Schunck- und Gaschutzklassen”, Mathematische Zeitschrift, 186 (1984), 149–178 (in German) | DOI

[11] Vasilyeva T. I., “On the intersections of generalized projectors with the products of normal subgroups of finite groups”, Problems of Physics, Mathematics and Technics, 2019, no. 2 (39), 61–65 (in Russian)

[12] Monakhov V. S., Introduction to the theory of finite groups and their classes, Vysheyshaya shkola, Minsk, 2006, 207 pp. (in Russian)

[13] Kamornikov S. F., “On formation subgroups of finite groups”, Arithmetic and Subgroup Structure of Finite Groups, Work of the Gomel Seminar, Nauka i tekhnika, Minsk, 1986, 69–74 (in Russian)

[14] Vedernikov V. A., Sorokina M. M., “$\omega$-fibered formations and Fitting classes of finite groups”, Mathematical Notes, 71:1 (2002), 39–55 | DOI | DOI

[15] Skiba A. N., Shemetkov L. A., “Multiple $\omega$-local formations and Fitting classes of finite groups”, Siberian Advances in Mathematics, 10:2 (2000), 112–141

[16] Vedernikov V. A., Sorokina M. M., “On properties of $\frak F^{\omega}$-projectors and $\frak F^{\omega}$-covering subgroups of finite groups”, Actual Problems of Applied Mathematics and Physics (May 17–21, 2017, Nalchik – Terskol), IAMA KBSC RAS, Nalchik, 2017, 262–263 (in Russian)

[17] Chunikhin S. A., Subgroups of finite groups, Nauka i tekhnika, Minsk, 1964, 158 pp. (in Russian)