Subgroups of the Fan of Sylow Subgroups and the Supersolvability of a Finite Group

T. I. Vasilyeva

Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 192-203.

Résumé

The notion of the fan of a subgroup of a group, which was introduced in 1979 by Z. I. Borevich, is used to prove the supersolvability of finite groups. It is proved that a finite group $G$ is supersolvable if and only if any basic subgroup of the fan of every Sylow subgroup either coincides with $G$ or can be connected with $G$ by a chain of subgroups with prime indices. We also prove the supersolvability of a finite group with supersolvable basic subgroups of the fan of every Sylow subgroup of the group.

Détail

Keywords: finite group, $\mathbb{P}$-subnormal subgroup, Sylow subgroup, basic subgroup of a fan, contranormalizer of a subgroup
Mots-clés : fan, supersolvable group.

Bibliographie

[1] Z. I. Borevich, “O raspolozhenii podgrupp”, Koltsa i moduli. 2, Zap. nauchn. sem. LOMI, 94, Izd-vo «Nauka», Leningrad. otd., L., 1979, 5–12 | MR | Zbl

[2] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss., 134, Springer-Verlag, Berlin, 1967 | MR | Zbl

[3] A. F. Vasilev, T. I. Vasileva, V. N. Tyutyanov, “O konechnykh gruppakh sverkhrazreshimogo tipa”, Sib. matem. zhurn., 51:6 (2010), 1270–1281 | MR

[4] L. S. Kazarin, “O gruppakh s faktorizatsiei”, Dokl. AN SSSR, 256:1 (1981), 26–29 | MR | Zbl

[5] A. F. Vasilev, T. I. Vasileva, V. N. Tyutyanov, “O proizvedeniyakh $\mathbb P$-subnormalnykh podgrupp v konechnykh gruppakh”, Sib. matem. zhurn., 53:1 (2012), 59–67 | MR

[6] V. N. Kniahina, V. S. Monakhov, “On supersolvability of finite groups with $\mathbb{P}$-subnormal subgroups”, Int. J. Group Theory, 2:4 (2013), 21–29 | MR | Zbl

[7] A. F. Vasilev, T. I. Vasileva, V. N. Tyutyanov, “O $\mathrm K$-$\mathbb P$-subnormalnykh podgruppakh konechnykh grupp”, Matem. zametki, 95:4 (2014), 517–528 | DOI | MR | Zbl

[8] A. Ballester-Bolinches, L. M. Ezquerro, A. A. Heliel, M. M. Al-Shomrani, “Some results on products of finite groups”, Bull. Malays. Math. Sci. Soc., 40 (2017), 1341–1351 | DOI | MR | Zbl

[9] A. Ballester-Bolinches, W. M. Fakieh, M. C. Pedraza-Aguilera, “On Products of Generalized Supersoluble Finite Groups”, Mediterr. J. Math., 16:46 (2019) | DOI | MR | Zbl

[10] A. Ballester-Bolinches, Y. Li, M. C. Pedraza-Aguilera, Ning Su, “Factorised Finite Groups”, Mediterr. J. Math., 17:65 (2020) | DOI | MR | Zbl

[11] V. S. Monakhov, A. A. Trofimuk, “On the residual of a factorized group with widely supersoluble factors”, Comm. Algebra, 48:12 (2020), 5290–5295 | MR | Zbl

[12] V. A. Vasilev, “O vliyanii submodulyarnykh podgrupp na stroenie konechnykh grupp”, Vesn. Vitsebsk. dzyarzh. un-ta, 91:2 (2016), 17–21

[13] I. Zimmermann, “Submodular subgroups in finite groups”, Math. Z., 202 (1989), 545–557 | DOI | MR | Zbl

[14] M. G. Bianchi, A. Gillio Berta Mayri, P. Hauck, “On finite soluble groups with nilpotent Sylow normalizers”, Arch. Math., 47 (1986), 193–197 | DOI | MR | Zbl

[15] V. Fedri, L. Serena, “Finite soluble groups with supersoluble Sylow normalizers”, Arch. Math., 50 (1988), 11–18 | DOI | MR | Zbl

[16] R. A. Bryce, V. Fedri, L. Serena, “Bounds on the Fitting length of finite soluble groups with supersoluble Sylow normalizers”, Bull. Austral. Math. Soc., 44 (1991), 19–31 | DOI | MR | Zbl

[17] A. Ballester-Bolinshe, L. A. Shemetkov, “O normalizatorakh silovskikh podgrupp v konechnykh gruppakh”, Sib. matem. zhurn., 40:1 (1999), 3–5 | MR | Zbl

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

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

[20] J. S. Rose, “Nilpotent subgroups of finite soluble groups”, Math. Z., 106 (1968), 97–112 | DOI | MR | Zbl

[21] A. F. Vasilev, “O proizvedeniyakh obobschenno abnormalnykh sverkhrazreshimykh podgrupp konechnykh grupp”, PFMT, 2:43 (2020), 52–58 | MR

[22] V. A. Vasilev, “Konechnye gruppy s submodulyarnymi silovskimi podgruppami”, Sib. matem. zhurn., 56:6 (2015), 1277–1288 | DOI | MR