Geodesic
On radicals of factorized finite $\mathfrak{X}$-groups
Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 69-75.

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Let $\mathfrak{X}$ be a saturated $S$-closed formation of finite soluble groups containing the class $\mathfrak{N}^k$ of all groups whose nilpotent length does not exceed $k$, where $k\geqslant 3$. In this paper, we obtain a constructive description of all Fitting formations $\mathfrak{F}$ in $\mathfrak{X}$ such that for any $\mathfrak{X}$-group $G = AB$, there is $A_{\mathfrak{F}}\cap B_{\mathfrak{F}}\subseteq G_{\mathfrak{F}}$.
Keywords: finite group, nilpotent length, di-$\mathfrak{F}$-group, $\mathfrak{F}$-radical, Fitting formation, radical formation with the Monakhov condition in the class $\mathfrak{X}$. 
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A. F. Vasil'ev. On radicals of factorized finite $\mathfrak{X}$-groups. Problemy fiziki, matematiki i tehniki, no. 4 (2021), pp. 69-75. http://geodesic.mathdoc.fr/item/PFMT_2021_4_a10/

[1] A.F. Vasilev, T.I. Vasileva, “O konechnykh gruppakh, u kotorykh glavnye faktory yavlyayutsya prostymi gruppami”, Izvestiya vuzov. Seriya Matematika, 426:11 (1997), 10–14

[2] V.S. Monakhov, “O trizhdy faktorizuemykh gruppakh”, Izvestiya akademii nauk BSSR. Seriya fiz.-mat. nauk, 1981, no. 6, 18–22

[3] P.M. Johnson, “A property of factorisable groups”, Arch. Math., 60 (1993), 414–419 | DOI

[4] B. Amberg, L.S. Kasarin, “On finite products of soluble groups”, Israel J. Math., 106 (1999), 93–108 | DOI

[5] A.F. Vasilev, “On radicals of products of finite soluble groups”, Voprosy algebry, 2000, no. 3 (16), 43–47

[6] L.A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978, 272 pp.

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

[8] I.N. Khalimonchik, “Formatsii Shemetkova v klasse $\mathfrak{X}$”, Izvestiya Gomelelskogo gosudarstvennogo universiteta imeni F. Skoriny, 55:4 (2009), 219–223

[9] V.D. Mazurov, E.I. Khukhro (sost.), Kourovskaya tetrad. Nereshennye voprosy teorii grupp, Inctitut matematiki im. S.L. Soboleva SO RAN, 14-e izd., Institut matematiki im. S.L. Soboleva SO RAN, Novosibirsk, 1999, 135 pp.

[10] E.A. Pennington, “On products of finite nilpotent groups”, Math. Z., 134 (1973), 81–83 | DOI | MR | Zbl

[11] H. Heineken, “Products of finite nilpotent groups”, Mat. Ann., 287 (1990), 643–652 | DOI | Zbl

[12] A.F. Vasilev, “O peresechenii faktorov konechnykh razreshimykh proizvedenii grupp”, Trudy Instituta matematiki, 14:2 (2006), 20–27

[13] B. Amberg, S. Franciosi, F. De Giovanni, Products of Groups, Oxford, 1992, 220 pp. | Zbl

[14] H. Heineken, “The products of two finite nilpotent groups its Fitting series”, Arch. Math., 59 (1992), 209–214 | DOI | MR | Zbl

[15] B. Amberg, B. Höfling, “On finite products of nilpotent groups”, Arch. Math., 63 (1994), 1–8 | DOI | Zbl

[16] A.F. Vasilev, “O faktorizuemykh in'ektorakh konechnykh razreshimykh grupp”, Vesnik Vitsebskaga dzyarzhaúnaga universiteta, 2005, no. 4 (38), 108–111

[17] A.F. Vasilev, “O faktorizatorakh podgrupp konechnykh razreshimykh di-$\mathfrak{F}$-grupp”, Vestsi Nats. akad. navuk Belarusi. Seryya fiz.-mat. navuk, 2006, no. 4, 55–60

[18] T.I. Vasileva, E.A. Ryabchenko, “O svoistvakh podgrupp konechnykh $\pi$-razreshimykh di-$\pi$-razlozhimykh grupp”, Vestsi Nats. akad. navuk Belarusi. Seryya fiz.-mat. navuk, 2009, no. 4, 15–19

[19] A. Ballester-Bolinches, R. Esteban-Romero, M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin, 2010, 334 pp. | Zbl