Finite groups of Lie type form the main array of finite simple groups, serving as a typical model for their classification, and have close relations with other areas of mathematics. An important class of local subgroups of finite groups of Lie type are their parabolic subgroups. Let $U$ be a normal subgroup of a finite group $P$. For a chief series of $P$, containing $U$ as a member, the normal series of subgroups of $U$ formed by all the members of the original series contained in $U$ is called the fragment of the original chief series of $P$ included in $U$. The determination of all fragments of chief series of $P$ included in $U$ implies the determination of all normal subgroups of $P$ contained in $U$. Let $G$ be a finite simple group of Lie type over a field of characteristic $p$, different from the Tits group ${}^2F_4(2)'$, and $P$ be a parabolic subgroup in $G$ with the unipotent radical $U$. The group $G$ is called special, if $p=2$ for $G$ of type $C_l$, $G_2$, $F_4$, ${}^2B_2$ or ${}^2F_4$ and $p=3$ for $G$ of type $G_2$ or ${}^2G_2$. The problem of determination of all fragments of chief series of $P$ included in $U$ for all parabolic maximal subgroups $P$ of $G$ was well studied in the case of non-special groups $G$ by several authors. The case of special groups $G$ is of significant interest, but was less studied. In this paper, we complete the determination of all fragments of chief series of $P$ included in $U$ and find the lower and the upper central series of $U$ for all special groups $G$ of exceptional Lie type and all their parabolic maximal subgroups $P$. Furthermore, similar results for the Tits group ${}^2F_4(2)'$ and all its 2-local maximal subgroups are obtained.
Keywords: finite simple group of Lie type, parabolic maximal subgroup, chief series, unipotent radical, fragment of chief series
Received May 14, 2025
Revised October 10, 2025
Accepted October 13, 2025
Published online December 24, 2025
Funding Agency: The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2025-1549).
Anatoly Semenovich Kondrat’ev, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Ural Mathematical Center, Yekaterinburg, 620077 Russia, е-mail: a.s.kondratiev@imm.uran.ru
Vera Vladimirovna Korableva, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620077 Russia; Prof., Chelyabinsk State University Chelyabinsk, 454001 Russia, е-mail: vvkora@gmail.com
Vladimir Ivanovich Trofimov, Dr. Phys.-Math. Sci., Lead. Sci. Researcher, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences; Ural Mathematical Center; Prof., Ural Federal University, Yekaterinburg, 620000 Russia, е-mail: trofimov@imm.uran.ru
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https://doi.org/10.1134/S003744661706012X
Cite this article as: A.S. Kondrat’ev, V.V. Korableva, V.I. Trofimov. On chief series of parabolic maximal subgroups of finite simple groups of exceptional Lie type. Trudy Instituta Matematiki i Mekhaniki URO RAN, 2026б vol. 32, no. 1, pp. 105–121.
