Nonpronormal subgroups of odd index in finite simple linear and unitary groups

W. Guo ; N. V. Maslova ; D. O. Revin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 1, pp. 70-79

Résumé

A subgroup $H$ of a group $G$ is pronormal if, for each $g \in G$, the subgroups $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$. Most of finite simple groups possess the following property $(*)$: each subgroup of odd index is pronormal in the group. The conjecture that all finite simple groups possess the property $(*)$ was established in 2012 in a paper by E. P. Vdovin and the third author based on the analysis of the proof that Hall subgroups are pronormal in finite simple groups. However, the conjecture was disproved in 2016 by A. S. Kondrat'ev together with the second and third authors. In a series of papers by Kondrat'ev and the authors published from 2015 to 2020, the finite simple groups with the property $(*)$ except finite simple linear and unitary groups with some constraints on natural arithmetic parameters were classified. In this paper we construct series of examples of nonpronormal subgroups of odd indices in finite simple linear and unitary groups over a field of odd characteristic, thereby making a step towards completing the classification of finite simple groups with the property $(*)$.

Détail

Keywords: finite group, linear simple group, unitary simple group, pronormal subgroup, odd index.
Mots-clés : simple group

Bibliographie

[1] Hall P., Phillip Hall's lecture notes on group theory, Part 6, University of Cambridge, Cambridge, 1951–1967 URL: http://omeka.wustl.edu/omeka/items/show/10788

[2] Babai L., “Isomorphism Problem for a Class of Point-Symmetric Structures”, Acta Math. Acad. Sci. Hungar., 29 (1977), 329–336 | DOI | MR | Zbl

[3] Palfy P.P., “Isomorphism Problem for Relational Structures with a Cyclic Automorphism”, Europ. J. Combinatorics, 8:1 (1987), 35–43 | DOI | MR | Zbl

[4] Praeger Ch.E., “On transitive permutation groups with a subgroup satisfying a certain conjugacy condition”, J. Austral. Math. Soc., 36:1 (1984), 69–86 | DOI | MR | Zbl

[5] Go V., Maslova N.V., Revin D.O., “O pronormalnosti podgrupp nechetnykh indeksov v nekotorykh rasshireniyakh konechnykh grupp”, Sib. mat. zhurn., 59:4 (2017), 773–790 | DOI

[6] de Giovanni F., Trombetti M., “Pronormality in group theory”, Adv. Group Theory Appl., 9 (2020), 123–149 | DOI | MR | Zbl

[7] Brescia M., Ferrara M., Trombetti M., “Groups whose subgroups are either abelian or pronormal ”, Kyoto J. Math., 63:3 (2023), 471–500 | DOI | MR | Zbl

[8] Brescia M., Trombetti M., “Locally finite simple groups whose non-Abelian subgroups are pronormal”, Comm. Algebra, 51:8 (2023), 3346–3353 | DOI | MR | Zbl

[9] Ferrara M., Trombetti M., “Groups with many pronormal subgroups”, Bull. Austral. Math. Soc., 105:1 (2022), 75–86 | DOI | MR | Zbl

[10] Ferrara M., Trombetti M., “Locally finite simple groups whose nonnonilpotent subgroups are pronormal”, publ. online, Bull. Austral. Math. Soc., 2023, 1–10 | DOI | MR

[11] Ferrara M., Trombetti M., “Periodic linear groups in which permutability is a transitive relation”, Ann. Mat. Pura Appl. (4), 203:1 (2024), 361–383 | DOI | MR

[12] Vdovin E. P., Revin D.O., “Pronormalnost khollovykh podgrupp v konechnykh prostykh gruppakh”, Sib. mat. zhurn., 53:3 (2012), 527–542 | MR | Zbl

[13] Kondratev A.S., Maslova N.V., Revin D.O., “O pronormalnosti podgrupp nechetnogo indeksa v konechnykh prostykh gruppakh”, Sib. mat. zhurn., 56:6 (2015), 1375–1383 | DOI | MR | Zbl

[14] Kondratev A.S., Maslova N.V., Revin D.O., “Kriterii pronormalnosti dobavlenii k abelevym normalnym podgruppam”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22:1 (2016), 153–158

[15] Kondratev A.S., Maslova N.V., Revin D.O., “O pronormalnosti podgrupp nechetnykh indeksov v konechnykh prostykh simplekticheskikh gruppakh”, Sib. mat. zhurn., 58:3 (2017), 599–610 | DOI | MR | Zbl

[16] Kondratev A.S., Maslova N.V., Revin D.O., “O pronormalnykh podgruppakh v konechnykh prostykh gruppakh”, Dokl. RAN, 482:1 (2018), 7–11 | DOI | Zbl

[17] Kondrat'ev A.S., Maslova N.V., Revin D.O., “Finite simple exceptional groups of Lie type in which all the subgroups of odd index are pronormal”, J. Group Theory, 23 (2020), 999–1016 | DOI | MR | Zbl

[18] Kondrat'ev A.S., Maslova N.V., Revin D.O., “On the pronormality of subgroups of odd index in finite simple groups”, Groups St Andrews 2017 in Birmingham, London Math. Soc. Lecture Note Ser., 455, eds. C.M. Campbell, M.R. Quick, C.W. Parker, E.F. Robertson, C.M. Roney-Dougal, Cambridge Univ. Press, Cambridge, 2019, 406–418 | DOI | MR | Zbl

[19] Maslova N.V., Revin D.O., “On the pronormality of subgroups of odd index in some direct products of finite groups”, J. Algebra Appl., 22:04 (2023), 2350083 | DOI | MR | Zbl

[20] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups. Number 3, Math. Surv. Monogr., 40, no. 3, 1994, 419 pp. | DOI | MR

[21] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, Cambridge University Press, Cambridge, 1990, 303 pp. | DOI | MR | Zbl

[22] Maslova N.V., “Klassifikatsiya maksimalnykh podgrupp nechetnogo indeksa v konechnykh prostykh klassicheskikh gruppakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:4 (2008), 100–118

[23] Maslova N.V., “Classification of maximal subgroups of odd index in finite simple classical groups: Addendum”, Sib. Electron. Math. Reports, 15 (2018), 707–718 | DOI | MR | Zbl

[24] Maslova N.V., “Maksimalnye podgruppy nechetnogo indeksa v konechnykh gruppakh s prostym lineinym, unitarnym ili simplekticheskim tsokolem”, Algebra i logika, 50:2 (2011), 189–208 | MR | Zbl

[25] Kondratev A.S., “Normalizatory silovskikh 2-podgrupp v konechnykh prostykh gruppakh”, Mat. zametki, 78:3 (2005), 368–376 | DOI | MR | Zbl