On Pronormality of Second Maximal Subgroups in Finite Groups with Socle L2(q)

Abstract

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 Hg are conjugate in hH, Hgi. 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 hH, Hgi 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 L2(211) to this question. In the present paper, we provide necessary and sufficient conditions for the pronormality of the second maximal subgroups in the group L2(q). In addition, for q ≤ 11, we find the finite almost simple groups with socle L2(q) in which all second maximal subgroups are pronormal. © 2020 Sverre Raffnsoe. All rights reserved.