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 Mat. Zametki, 2001, Volume 69, Issue 4, Pages 524–549 (Mi mz521)

Maximal Orders of Abelian Subgroups in Finite Chevalley Groups

E. P. Vdovin

Novosibirsk State University

Abstract: In the present paper, for any finite group $G$ of Lie type (except for $^2F_4(q)$), the order $a(G)$ of its large Abelian subgroup is either found or estimated from above and from below (the latter is done for the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and $^2E_6(q^2)$). In the groups for which the number $a(G)$ has been found exactly, any large Abelian subgroup coincides with a large unipotent or a large semisimple Abelian subgroup. For the groups $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$ and $^2E_6(q^2)$, it is shown that if an Abelian subgroup contains a noncentral semisimple element, then its order is less than the order of an Abelian unipotent group. Hence in these groups the large Abelian subgroups are unipotent, and in order to find the value of $a(G)$ for them, it is necessary to find the orders of the large unipotent Abelian subgroups. Thus it is proved that in a finite group of Lie type (except for $^2F_4(q)$) any large Abelian subgroup is either a large unipotent or a large semisimple Abelian subgroup.

DOI: https://doi.org/10.4213/mzm521

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English version:
Mathematical Notes, 2001, 69:4, 475–498

Bibliographic databases:

UDC: 512.542.5
Revised: 01.10.2000

Citation: E. P. Vdovin, “Maximal Orders of Abelian Subgroups in Finite Chevalley Groups”, Mat. Zametki, 69:4 (2001), 524–549; Math. Notes, 69:4 (2001), 475–498

Citation in format AMSBIB
\Bibitem{Vdo01} \by E.~P.~Vdovin \paper Maximal Orders of Abelian Subgroups in Finite Chevalley Groups \jour Mat. Zametki \yr 2001 \vol 69 \issue 4 \pages 524--549 \mathnet{http://mi.mathnet.ru/mz521} \crossref{https://doi.org/10.4213/mzm521} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1845994} \zmath{https://zbmath.org/?q=an:0994.20013} \transl \jour Math. Notes \yr 2001 \vol 69 \issue 4 \pages 475--498 \crossref{https://doi.org/10.1023/A:1010256129959} 

• http://mi.mathnet.ru/eng/mz521
• https://doi.org/10.4213/mzm521
• http://mi.mathnet.ru/eng/mz/v69/i4/p524

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This publication is cited in the following articles:
1. Vdovin, EP, “The number of subgroups with trivial unipotent radicals in finite groups of Lie type”, Journal of Group Theory, 7:1 (2004), 99
2. V. M. Levchuk, G. S. Suleimanova, “Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups”, Proc. Steklov Inst. Math. (Suppl.), 267, suppl. 1 (2009), S118–S127
3. Ahanjideh, N, “A CHARACTERIZATION OF Bn(q) BY THE SET OF ORDERS OF MAXIMAL ABELIAN SUBGROUPS”, International Journal of Algebra and Computation, 19:2 (2009), 191
4. Vladimir M. Levchuk, Galina S. Suleimanova, “Thompson subgroups and large abelian unipotent subgroups of Lie-type groups”, Zhurn. SFU. Ser. Matem. i fiz., 6:1 (2013), 63–73
5. Yuri Prokhorov, Constantin Shramov, “Jordan constant for Cremona group of rank $3$”, Mosc. Math. J., 17:3 (2017), 457–509
6. Kirillova E.A., Suleimanova G.S., “Highest Dimension Commutative Ideals of a Niltriangular Subalgebra of a Chevalley Algebra Over a Field”, Tr. Inst. Mat. Mekhaniki URO RAN, 24:3 (2018), 98–108
7. Suleimanova G.S., “The Highest Dimension of Commutative Subalgebras in Chevalley Algebras”, J. Sib. Fed. Univ.-Math. Phys., 12:3 (2019), 351–354
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