Injectors of finite $\sigma$-soluble groups

N. T. Vorob'ev ; E. D. Volkova

Problemy fiziki, matematiki i tehniki, no. 1 (2023), pp. 75-84.

Résumé

Let $\sigma=\{\sigma_i: i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, i. e. $\mathbb{P}=\cup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$. Finite group $G$ is $\sigma$-soluble, if every chief factor $H/K$ of $G$ is a $\sigma_i$-group for some $\sigma_i\in\sigma$. Fitting class $\mathfrak{H}=\cap_{\sigma_i\in\sigma}h(\sigma_i)\mathfrak{E}_{\sigma_i'}\mathfrak{E}_{\sigma_i}$ is said to be $\sigma$-class Hartley. In this paper we prove the existence and conjugacy of $\mathfrak{H}$-injectors of $G$ and describe their characterization in the terminal of the radicals.

Détail

Mots-clés : $\sigma$-soluble group
Keywords: $\sigma$-class Hartley, injector.

Bibliographie

[1] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New-York, 1992, 891 pp. | MR

[2] M.L. Sylow, “Theoremes sur les groupes de substitutions”, Math. Ann., 5 (1872), 584–594 | DOI | MR

[3] P. Hall, “A note on soluble groups”, J. London Math. Soc., 3 (1928), 98–105 | DOI | MR

[4] B. Fisher, W. Gaschütz, B. Hartley, “Injektoren endlicher auflosbarer Gruppen”, Math. Z., 102 (1967), 337–339 | DOI | MR

[5] B. Fisher, Klassen konjugierter Untergruppen in endlichen auflösbaren Grouppen, Habilitationsschreft, Universität Frankfurt am Mainz, 1966

[6] B. Hartley, “On Fischer's dualization of formation theory”, Proc. London Math. Soc., 3:2 (1969), 193–207 | DOI | MR | Zbl

[7] D. Blessenohl, H. Laue, “Fittingklassen endlicher Gruppen in denen gewisse Haupfaktoren einfach sind”, J. Algebra, 56:3 (1979), 516–532 | DOI | MR | Zbl

[8] L.A. Shemetkov, “O podgruppakh $\pi$-razreshimykh grupp”, Konechnye gruppy, Nauka i tekhnika, Minsk, 1975, 207–212

[9] L.A. Shemetkov, “Nekotorye svoistva in'ektorov v konechnykh gruppakh”, Izvestiya Gomelskogo gosudarstvennogo universiteta imeni F. Skoriny. Voprosy algebry, 1999, no. 1 (15), 5–13

[10] L.A. Shemetkov, “Injectors in finite groups”, Izvestija Gomel'skogo gos. un-ta im. F. Skoriny. Voprosy algebry, 2000, no. 3 (16), 186–187 | Zbl

[11] Y.F. Liu, W. Guo, N.T. Vorob'ev, “Description of $\mathfrak{F}$-injectors of finite soluble groups”, Math. Sci. Res. J., 12:1 (2008), 17–22 | MR | Zbl

[12] W. Guo, N.T. Vorob'ev, “On injectors of finite soluble groups”, Comm. Algebra, 36 (2008), 3200–3208 | DOI | MR | Zbl

[13] N. Yang, W. Guo, N.T. Vorob'ev, “On $\mathcal{F}$-injectors of Fitting set of a finite group”, Communications in Algebra, 46:1 (2018), 217–229 | DOI | MR | Zbl

[14] A.N. Skiba, “A generalization of a Hall theorem”, J. Algebra and Appl., 15:5 (2015), 21–36 | MR

[15] A.N. Skiba, “On $\sigma$-properties of Finite groups II”, Problems of Physics, Mathematics and Technics, 2015, no. 3 (24), 70–83 | MR | Zbl

[16] Chzhan Chi, A.N. Skiba, “O $\Sigma_t^\sigma$-zamknutykh klassakh konechnykh grupp”, Ukrainskii matematicheskii zhurnal, 70:12 (2018), 1707–1716

[17] N.T. Vorobev, “O radikalnykh klassakh konechnykh grupp s usloviem Loketta”, Matematicheskie zametki, 43:2 (1988), 161–168 | Zbl

[18] L.A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978

[19] M.J. Iranzo, P. Monasor, “$\mathfrak{F}$-constraint with respect to a Fitting class”, Arch. Math., 46:5 (1986), 205–210 | DOI | MR | Zbl

[20] W. Guo, L. Zhang, N.T. Vorob'ev, “On $\sigma$-local Fitting classes”, J. Algebra, 542:15 (2020), 116–129 | DOI | MR | Zbl

[21] Zhang Chi, V.G. Safonov, A.N. Skiba, “On $n$-multiply $\sigma$-local formations of finite groups”, Communications in Algebra, 47:3 (2019), 957–968 | DOI | MR | Zbl

[22] N.T. Vorobev, T.B. Karaulova, “Mnozhestva Khartli i in'ektory konechnoi gruppy”, Matematicheskie zametki, 105:2 (2019), 214–227 | DOI | Zbl