Using R. Wilson's recent results, we prove the existence of triples $(\mathfrak{X},G,H)$ such that $\mathfrak{X}$ is a complete (i.e. closed under taking subgroups, homomorphic images, and extensions) class of finite groups, $G$ is a finite simple group, $H$ is an $\mathfrak{X}$-maximal subgroup of $G$, and $H$ is not pronormal in $G$. This disproves a conjecture stated earlier by the second author and W. Guo.
Keywords: complete class of groups, relatively maximal subgroup, pronormal subgroup, finite simple group
Received August 28, 2023
Revised September 14, 2023
Accepted September 18, 2023
Funding Agency: The research of B. Li was supported by National Natural Science Foundation of China, (NSFC), grant no.12371021. The research of D.O.Revin was carried out within the State Contract of the Sobolev Institute of Mathematics (FWNF-2022-0002).
Baojun Li, PhD, Prof., School of Sciences, Nantong University, Nantong, 226019 P.R. China, e-mail: libj@ntu.edu.cn
Danila Olegovich Revin, Dr. Phys.-Math. Sci., Prof., Sobolev Institute of Mathematics of the Siberia Branch of the Russian Academy of Sciences, Novosibirsk, 630090 Russia; Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: revin@math.nsc.ru
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Cite this article as: B. Li, D.O. Revin. Examples of non-pronormal relatively maximal subgroups in finite simple groups. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 140–145; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 323, Suppl. 1, pp. S155–S159.
