×

On K-\(\mathbb P\)-subnormal subgroups of finite groups. (English. Russian original) Zbl 1330.20033

Math. Notes 95, No. 4, 471-480 (2014); translation from Mat. Zametki 95, No. 4, 517-528 (2014).
Summary: A subgroup \(H\) of a group \(G\) is said to be K-\(\mathbb P\)-subnormal in \(G\) if \(H\) can be joined to the group by a chain of subgroups each of which is either normal in the next subgroup or of prime index in it. Properties of K-\(\mathbb P\)-subnormal subgroups are obtained. A class of finite groups whose Sylow \(p\)-subgroups are K-\(\mathbb P\)-subnormal in \(G\) for every \(p\) in a given set of primes is studied. Some products of K-\(\mathbb P\)-subnormal subgroups are investigated.

MSC:

20D35 Subnormal subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D30 Series and lattices of subgroups
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] L. A. Shemetkov, Formations of Finite Groups (Nauka, Moscow, 1978) [in Russian]. · Zbl 0496.20014
[2] A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, in Math. Appl. (Springer) (Dordrecht, Springer, 2006), Vol. 584. · Zbl 1102.20016
[3] O. H. Kegel, ”Untergruppenverbände endlicher Gruppen, die den Subnormalteilerverband echt enthalten,” Arch. Math. (Basel) 30(3), 225–228 (1978). · Zbl 0943.20500
[4] A. F. Vasil’ev, T. I. Vasil’eva, and V. N. Tyutyanov, ”On the finite groups of supersoluble type,” Sibirsk. Mat. Zh. 51(6), 1270–1281 (2010) [Sib. Math. J. 51 (6), 1004–1012 (2010)].
[5] A. F. Vasil’ev, T. I. Vasil’eva, and V. N. Tyutyanov, ”On the products of \(\mathbb{P}\)-subnormal subgroups of finite groups,” Sibirsk. Mat. Zh. 53(1), 59–67 (2012) [Sib. Math. J. 53 (1), 47–54 (2012)].
[6] A. F. Vasil’ev, T. I. Vasil’eva, and V. N. Tyutyanov, ”On finite groups similar to supersolable groups,” PFMT, No. 2, 21–27 (2010) [in Russian].
[7] V. N. Kniahina and V. S. Monakhov, Finite Groups with \(\mathbb{P}\)-Subnormal Primary Cyclic Subgroups, arXiv: math.GR/ 1110.4720v2 (2011).
[8] L. S. Kazarin, ”On factorizable groups,” Dokl. Akad. Nauk SSSR 256(1), 26–29 (1981) [Soviet Math. Dokl. 23 (1), 19–22 (1981)].
[9] K. Doerk and T. Hawkes, Finite Soluble Groups, in de Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol. 4. · Zbl 0753.20001
[10] A. F. Vasil’ev, ”New properties of finite dinilpotent groups,” Vestsi Nats. Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk, No. 2, 29–33 (2004) [in Russian].
[11] Between Nilpotent and Solvable, Ed. by M. Weinstein (Polygonal Publ. House, Washington, NJ, 1982).
[12] L. S. Kazarin, ”On the product of finite groups,” Dokl. Akad. Nauk SSSR 269(3), 528–531 (1983) [Soviet Math. Dokl. 27 (2), 354–357 (1983)].
[13] R. Baer, ”Classes of finite groups and their properties,” Illinois J. Math., No. 1, 115–187 (1957). · Zbl 0077.03003
[14] A. F. Vasil’ev and T. I. Vasil’eva, ”On finite groups in which the principal factors are simple groups,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 10–14 (1997) [Russian-Math. (Iz. VUZ) 41 (11), 8–12 (1997) (1998)].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.