We study $D_\pi$-groups with a unit solvable radical that do not have nontrivial normal $\pi$-subgroups in which all simple nonabelian factors of their subnormal series are simple sporadic groups. It is proved that in such groups, for any $\pi$-Hall subgroup $H$, there exists an element $g$ such that $H\cap H^g=1$. Thus, Question 20.123 (c) of the Kourovka Notebook is solved and, under the above conditions, a positive answer is given to Question 18.31.
Keywords: Hall subgroup, $D_\pi$-group
Received November 18, 2024
Revised January 23, 2025
Accepted January 27, 2025
Ivan Nikolaevich Belousov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, 620000 Russia, e-mail: i_belousov@mail.ru
Victor Ivanovich Zenkov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, 620000 Russia, e-mail: v1i9z52@mail.ru
REFERENCES
1. Zenkov V.I., Mazurov V.D. On the intersection of Sylow subgroups in finite groups. Algebra and Logic, 1996, vol. 35, no. 4, pp. 236–240. https://doi.org/10.1007/BF02367025
2. Zenkov V.I. The intersections of nilpotent subgroups in finite groups. Fundam. Prikl. Mat., 1996, vol. 2, no. 1, pp. 1–92 (in Russian).
3. Vdovin E.P. Regular orbits of solvable linear p′-groups. Sib. Elektron. Mat. Izv., 2007, vol. 4, pp. 345–360.
4. Dolfi S. Large orbits in coprime actions of solvable groups. Trans. Amer. Math. Soc., 2008, vol. 360, no. 1, pp. 135–152. https://doi.org/10.1090/S0002-9947-07-04155-4
5. Vdovin E.P., Revin D.O. Theorems of Sylow type. Russian Math. Surv., 2011, vol. 66, no. 5, pp. 829–870. https://doi.org/10.1070/RM2011v066n05ABEH004762
6. The Kourovka notebook. Unsolved problems in group theory / eds. V.D. Mazurov, E.I. Khukhro. 20th ed. [e-resource]. Linkoln, Novosibirsk, University of Lincoln, U.K.; Inst. Math. SO RAN, 2024, 274 p. Available at: https://kourovka-notebook.org/ .
7. Zenkov V.I. On intersections of $\pi$-Hall subgroups in finite $D_\pi$-groups. Sib. Math. J., 2022, vol. 63, no. 4, pp. 720-–722. https://doi.org/10.1134/S0037446622040127
8. Revin D.O. The $D_\pi$-property in finite simple groups. Algebra and Logic, 2008, vol. 47, no. 3, pp. 210–227. https://doi.org/10.1007/s10469-008-9010-4
9. Revin D.O., Vdovin E.P. On the number of classes of conjugate Hall subgroups in finite simple groups. J. Algebra, 2010, vol. 324, no. 12, pp. 3614–3652. https://doi.org/ 10.1016/j.jalgebra.2010.09.014
10. Zenkov V.I. Intersections of nilpotent subgroups in finite groups with sporadic socle. Algebra and Logic, 2020, vol. 59, no. 4, pp. 313–321. https://doi.org/10.1007/s10469-020-09603-x
11. Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A. Atlas of finite groups. Oxford, Clarendon Press, 1985, 252 p. ISBN: 978-0-19-853199-9.
12. Gorenstein D. Finite simple groups. An introduction to their classification. New York, Springer, 1982, 333 p. https://doi.org/10.1007/978-1-4684-8497-7 . Translated to Russian under the title Konechnye prostye gruppy. Vvedenie v ikh klassifikatsiyu, Moscow, Mir Publ., 1985, 352 p.
13. Gorenstein D., Lyons R. The local structure of finite groups of characteristic 2 type. Mem. Amer. Math. Soc., vol. 42, edt. 276, Providence, RI, Am. Math. Soc., 1983, 731 p. https://doi.org/ 10.1090/memo/0276
14. Kabanov V.V., Kondrat’ev A.S. Silovskiye 2-podgruppy konechnykh grupp (obzor) [Sylow 2-subgroups of finite groups (Review)]. Sverdlovsk: Institute of Mathematics and Mechanics of the USSR Academy of Sciences Publ., 1979, 144 p. (Full text)
15. Gluck D. Trivial set-stabilizers in finite permutation groups. Canad. J. Math., 1983, vol. 35, no. 1, pp. 59–67. https://doi.org/10.4153/CJM-1983-005-2
16. Belousov I.N. I: Intersections $\pi$-Hall $D_\pi$-subgroups in finite simple sporadic groups, II: Finding a Lower Bound for Orbp(G) using the Sylow p-subgroup center centralizer. Available at: https://github. com/BelousovIN/Intersection_Subgroup , In: The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.14.0; 2024. (https://www.gap-system.org).
17. Vdovin E.P., Manzaev N.Ch., Revin D.O. On the heritability of the Sylow $\pi$-theorem by subgroups. Sb. Math., 2020, vol. 211, no. 3, pp. 309–335. https://doi.org/10.1070/SM9185
18. Halasi Z., Podoski K. Every coprime linear group admits a base of size two. Trans. Amer. Math. Soc., 2016, vol. 368, no. 8, pp. 5857–5887. https://doi.org/10.1090/tran/6544
19. Feit W., Thompson J.G. Solvability of groups of odd order. Pacific J. Math., 1963, vol. 13, no. 3, pp. 775–787. https://doi.org/10.2140/pjm.1963.13.775
20. Glauberman G. Factorizations in local subgroups of finite groups. In: CBMS Regional Conf. Ser. Math., vol. 33, Providence, R.I., Amer. Math. Soc., 1977. https://doi.org/10.1090/cbms/033
21. Kargapolov M.I., Merzlyakov Yu.I. Osnovy teorii grupp [Fundamentals of group theory], Moscow, Nauka Publ., 1982, 240 p.
22. Supplementary Materials for the article I. N. Belousov, V. I. Zenkov. “On intersections of $\pi$-Hall subgroups of some $D_\pi$-groups”. The online version contains supplementary material available at http://journal. imm.uran.ru/Suppl_inf_2025-v.31-1 (Full text) .
Cite this article as: I.N. Belousov, V.I. Zenkov. On intersections of $\pi$-Hall subgroups of some $D_\pi$-groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 19–35.
