Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/111253
Title: On Pronormality of Second Maximal Subgroups in Finite Groups with Socle L2(q)
Other Titles: О пронормальности вторых максимальных подгрупп в конечных группах с цоколем L2(q)
Authors: Zenkov, V. I.
Issue Date: 2020
Publisher: Krasovskii Institute of Mathematics and Mechanics
Krasovskii Institute of Mathematics and Mechanics UB RAS
Citation: Zenkov V. I. On Pronormality of Second Maximal Subgroups in Finite Groups with Socle L2(q) [О пронормальности вторых максимальных подгрупп в конечных группах с цоколем L2(q)] / V. I. Zenkov // Trudy Instituta Matematiki i Mekhaniki UrO RAN. — 2020. — Vol. 26. — Iss. 3. — P. 32-43.
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.
Keywords: FINITE GROUP
MAXIMAL SUBGROUP
PRONORMAL SUBGROUP
SIMPLE GROUP
URI: http://hdl.handle.net/10995/111253
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85095700272
PURE ID: 13944929
ISSN: 0134-4889
metadata.dc.description.sponsorship: 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).
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