V.I. Zenkov. On pronormality of second maximal subgroups in finite groups with socle $L_2(q)$ ... P. 32-43

According to Ph. Hall, a subgroup $H$ of a finite group $G$ is called pronormal in $G$ if, for any element $g$ of $G$, the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$. The simplest examples of pronormal subgroups of finite groups are normal subgroups, maximal subgroups, and Sylow subgroups. Pronormal subgroups of finite groups were studied by a number of authors. For example, Legovini (1981) studied finite groups in which every subgroup is subnormal or pronormal. Later, Li and Zhang (2013) described the structure of a finite group $G$ in which, for a second maximal subgroup $H$, its index in $\langle H,H^g\rangle$ does not contain squares for any $g$ from $G$. A number of papers by Kondrat'ev, Maslova, Revin, and Vdovin (2012-2019) are devoted to studying the pronormality of subgroups in a finite simple nonabelian group and, in particular, the existence of a nonpronormal subgroup of odd index in a finite simple nonabelian group. In the Kourovka Notebook, the author formulated Question 19.109 on the equivalence in a finite simple nonabelian group of the condition of pronormality of its second maximal subgroups and the condition of Hallness of its maximal subgroups. Tyutyanov gave a counterexample $L_2(2^{11})$ to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of the second maximal subgroups in the group $L_2(q)$. In addition, for $q\le 11$, we find the finite almost simple groups with socle $L_2(q)$ in which all second maximal subgroups are pronormal.

Keywords: finite group, simple group, maximal subgroup, pronormal subgroup

Received October 29, 2019

Revised July 11, 2020

Accepted August 3, 2020

Funding Agency: This work was supported 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, Yekaterinburg, 620002 Russia, e-mail: v1i9z52@mail.ru

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Cite this articlr as: V.I. Zenkov. On pronormality of second maximal subgroups in finite groups with socle $L_2(q)$, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 32–43; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 315, Suppl. 1, pp. S250–S260.