V.I. Zenkov. On intersections of nilpotent subgroups in finite groups with simple socle from the Atlas ... P. 54-66

Earlier, the author described up to conjugation all pairs $(A,B)$ of nilpotent subgroups of a finite group $G$ with socle $L_2(q)$ for which $A\cap B^g\ne 1$ for any element of $G$. A similar description was obtained by the author later for primary subgroups $A$ and $B$ of a finite group $G$ with socle $L_n(2^m)$. In this paper, we describe up to conjugation all pairs $(A,B)$ of nilpotent subgroups of a finite group $G$ with simple socle from the "Atlas of Finite Groups" for which $A\cap B^g\ne 1$ for any element $g$ of $G$. The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the "Kourovka Notebook") that a finite simple non-Abelian group $G$ for any nilpotent subgroups $N$ contains an element $g$ such that $N\cap N^g=1$.

Keywords: finite group, nilpotent subgroup, intersection of subgroups, Fitting subgroup

Received April 22, 2022

Revised April 21, 2023

Accepted May 15, 2023

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00456) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013,  between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

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

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Cite this article as: V.I. Zenkov. On intersections of nilpotent subgroups in finite groups with the simple socle from Atlas. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 2, pp. 54–66; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S321–S332.