On Products of $\mathrm{F}^*(G)$-Subnormal Subgroups of Finite Groups

V. I. Murashka

Matematičeskie zametki, Tome 111 (2022) no. 2, pp. 277-286.

Résumé

A subgroup $H$ of a finite group $G$ is said to be $\mathrm{F}^*(G)$-subnormal if it is subnormal in $H\mathrm{F}^*(G)$, where $\mathrm{F}^*(G)$ is the generalized Fitting subgroup of $G$. In the paper, the structure of groups factorizable by two $\mathrm{F}^*(G)$-subnormal subgroups, one of which is nilpotent, is studied. In particular, if the other factor is metanilpotent, then the group is solvable. Moreover, if the commutator subgroup of the second factor is nilpotent, then the nilpotent length of the group is at most 3. The supersolvability of the product $G=AB=AC=BC$ of a nilpotent subgroup $A$ by supersolvable subgroups $B$ and $C$ (all three are $\mathrm{F}^*(G)$-subnormal) is established.

Détail

Keywords: finite group, nilpotent group, generalized Fitting subgroup, $\mathrm{F}^*(G)$-subnormal subgroup.
Mots-clés : solvable group

Bibliographie

[1] H. Fitting, “Beiträge zur Theorie der endlichen Gruppen”, Jahresber. Deutsch. Math.-Verein., 48 (1938), 77–141

[2] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of Finite Groups, De Gruyter, Berlin, 2010 | MR | Zbl

[3] Between Nilpotent and Solvable, ed. M. Weinstein, Polygonal Publ. House, Washington, NJ, 1982 | Zbl

[4] O. H. Kegel, “Zur Struktur mehrfach faktorisierbarer endlicher Gruppen”, Math. Z., 87:1 (1965), 42–48 | DOI | Zbl

[5] M. Asaad, A. Shaalan, “On the supersolubility of finite groups”, Arch. Math., 53 (1989), 318–326 | DOI | Zbl

[6] A. Ballester-Bolinches, J. Cossey, M. C. Pedraza-Aguliera, “On products of finite supersoluble groups”, Comm. Algebra, 27:7 (2001), 3145–3152 | DOI

[7] V. S. Monakhov, A. A. Trofimuk, “On products of finite w-supersoluble groups”, Mediterr. J. Math., 18 (2021), Paper No. 100 | DOI | Zbl

[8] V. S. Monakhov, I. K. Chirik, “O sverkhrazreshimom koradikale proizvedeniya subnormalnykh sverkhrazreshimykh podgrupp”, Sib. matem. zhurn., 58:2 (2017), 353–364 | DOI | Zbl

[9] V. I. Murashko, A. V. Vasilev, “O proizvedeniyakh chastichno subnormalnykh podgrupp konechnykh grupp”, Vestn. VGU, 70:4 (2012), 24–27

[10] B. Huppert, N. Blackburn, Finite Groups III, Springer, Berlin, 1982 | MR | Zbl

[11] A. F. Vasilev, V. I. Murashko, “Formatsii i proizvedeniya $\mathrm{F}(G)$-subnormalnykh podgrupp konechnykh razreshimykh grupp”, Matem. Zametki, 107:3 (2020), 76–390 | DOI | MR

[12] V. S. Monakhov, I. K. Chirik, “Konechnye gruppy, faktorizuemye subnormalnymi sverkhrazreshimymi podgruppami”, PFMT, 2016, no. 3 (28), 40–46 | Zbl

[13] V. S. Monakhov, A. A. Trofimuk, “Konechnye gruppy faktorizuemye $T\mathbb{X}$-subnormalnymi podgruppami”, Ukr. matem. zhurn. (to appear)

[14] A. A. Trofimuk, Konechnye faktorizuemye gruppy, somnozhiteli kotorykh uslovno perestanovochny ili imeyut zadannoe vlozhenie, Izd. tsentr BGU, Minsk, 2020

[15] V. I. Murashko, “Proizvedeniya $\mathrm{F}^*(G)$-subnormalnykh podgrupp konechnykh grupp”, Izv. vuzov. Matem., 2017, no. 6, 76–82 | Zbl

[16] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin, 1992 | MR

[17] A. F. Vasilev, “O nekotorykh svoistvakh lokalnykh formatsii”, Voprosy algebry, Universitetskoe, Minsk, 1985, 4–9

[18] A. Ballester-Bolinches, L. M. Ezquerro, “On formations with the Kegel property”, J. Group Theory, 8 (2005), 605–611 | DOI | Zbl

[19] A. F. Vasilev, T. I. Vasileva, “O trizhdy faktorizuemykh konechnykh razreshimykh gruppakh”, Vesn. NAN Belarusi. Ser. fiz.-mat. navuk., 1997, no. 2, 36–39 | MR

[20] A. F. Vasilev, D. N. Simonenko, “Otnositelno radikalnye lokalnye formatsii”, Izv. Gomelskogo gos. un-ta im. F. Skoriny, 38:5 (2006), 19–25

[21] Xinjian Zhang, “A generalization of Baer's theorem”, Comm. Algebra, 45:11 (2017), 4971-4973 | DOI | Zbl

[22] A. Ballester-Bolinches, L. M. Ezquerro, Classes of Finite Groups, Springer, Dordrecht, 2006 | MR | Zbl

[23] L. A. Shemetkov, Formatsii konechnykh grupp, Sovremennaya algebra, Nauka, M., 1978 | MR | Zbl