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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, Volume 30, Number 1, Pages 70–79
DOI: https://doi.org/10.21538/0134-4889-2024-30-1-70-79
(Mi timm2063)
 

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

W. Guoab, N. V. Maslovacd, D. O. Revinec

a School of Mathematics and Statistics, Hainan University
b University of Science and Technology of China, Anhui, Hefei
c N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
d Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
e Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
References:
Abstract: 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 $(*)$.
Keywords: finite group, simple group, linear simple group, unitary simple group, pronormal subgroup, odd index.
Funding agency Grant number
Russian Science Foundation 19-71-10067
National Natural Science Foundation of China 12171126
12371021
Ministry of Science and Higher Education of the Russian Federation FWNF-2022-0002
This work was supported by the Russian Science Foundation (project no. 19-71-10067, Theorem 1), the National Natural Science Foundation of China (project nos. 12171126 and 12371021), and within a state contract of the Sobolev Institute of Mathematics (FWNF-2022-0002).
Received: 05.12.2023
Revised: 08.01.2024
Accepted: 15.01.2024
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2024, Volume 325, Issue 1, Pages S114–S122
DOI: https://doi.org/10.1134/S0081543824030088
Bibliographic databases:
Document Type: Article
UDC: 512.542
Language: Russian
Citation: W. Guo, N. V. Maslova, D. O. Revin, “Nonpronormal subgroups of odd index in finite simple linear and unitary groups”, Trudy Inst. Mat. i Mekh. UrO RAN, 30, no. 1, 2024, 70–79; Proc. Steklov Inst. Math. (Suppl.), 325, suppl. 1 (2024), S114–S122
Citation in format AMSBIB
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\by W.~Guo, N.~V.~Maslova, D.~O.~Revin
\paper Nonpronormal subgroups of odd index in finite simple linear and unitary groups
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2024
\vol 30
\issue 1
\pages 70--79
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\crossref{https://doi.org/10.21538/0134-4889-2024-30-1-70-79}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2024
\vol 325
\issue , suppl. 1
\pages S114--S122
\crossref{https://doi.org/10.1134/S0081543824030088}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85201972158}
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