On intersections of $\pi$-Hall subgroups of some $D_\pi$-groups

I. N. Belousov^ab, V. I. Zenkov<sup>ab

a</sup> N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: 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.

UDC: 512.542, 519.6

MSC: 20D10, 20B40

Received: 18.11.2024
Revised: 23.01.2025
Accepted: 27.01.2025

DOI: 10.21538/0134-4889-2025-31-1-19-35

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