Abstract
According to P. 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 second maximal subgroups in the group \(L_{2}(q)\). In addition, for \(q\leq 11\), we find the finite almost simple groups with socle \(L_{2}(q)\) in which all second maximal subgroups are pronormal.
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 3, pp. 32 - 43, 2020 https://doi.org/10.21538/0134-4889-2020-26-3-32-43.
Translated by E. Vasil’eva
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Zenkov, V.I. On the Pronormality of Second Maximal Subgroups in Finite Groups with Socle \(L_{2}(q)\). Proc. Steklov Inst. Math. 315 (Suppl 1), S250–S260 (2021). https://doi.org/10.1134/S0081543821060201
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DOI: https://doi.org/10.1134/S0081543821060201