Abstract
Automorphisms, isomorphisms and structural descriptions of finitary unitriangular groups over a ring and of Sylow subgroups of Chevalley groups are studied. Also, the theorem on pair unipotent intersections of Chevalley groups has been proved; it is connected with the known question on pair Sylow intersections of finite groups. These investigations use close connections and structural descriptions of considered groups and associated rings, which have been found recently in the case of a commutative ring of coefficients with a strongly maximal ideal. We consider some properties and examples of such ideals.
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References
Carter, R.: Simple Groups of Lie Type, Wiley, New York, 1972.
Hungerford, T. W.: Algebra, Winston, New York, 1974.
Kabanov, V. V. and Kondratyev, A. S.: Sylow 2-subgroups of finite groups, IMM Ural Sc. Cent. AN USSR, Sverdlovsk, 1979.
Kabanov, V. V., Makhnev, A. A. and Starostin, A. I.: Finite groups with normal intersections of Sylow 2-subgroups, Algebra i Logic 15(6) (1976), 655–659 (in Russian).
The Kourovka Notebook. Unsolved Problems in Group Theory, 12th edn, Inst. of Math., Sib. Dept. of Rus. Acad. Sci., Novosibirsk, 1992.
Kuzucuoglu, F. and Levchuk, V. M.: Ideals of some matrix rings, Comm. Algebra 28(7) (2000), 3503–3513.
Kuzucuoglu, F. and Levchuk, V. M.: The automorphism group of certain radical matrix rings, J. Algebra 243(2) (2001), 473–485.
Levchuk, V. M.: Connections between the unitriangular group and rings. II. Groups of automorphisms, Sibirsk. Mat. Zh. 24(4) (1983), 543–557.
Levchuk, V. M.: Some locally nilpotent matrix rings, Mat. Zametki 42(5) (1987), 631–641.
Levchuk, V. M.: Automorphisms of unipotent subgroups of Chevalley groups, Algebra and Logic 29(3) (1990), 315–338.
Levchuk, V. M.: Chevalley groups and their unipotent subgroups, Contemp. Math. 131(1) (1992), 227–242.
Levchuk, V. M.: Hypercentral series and pair intersections of Sylow subgroups of Chevalley groups, Algebra i Logic 41(6) (2002), 530–544.
Levchuk, V. M. and Suleimanova, G. S.: A normal structure of unipotent subgroup of Steinberg group over a field, Vestnik Krasnoyarsk St. Tekhn. Univ. 16 (1999), 44–48.
Levchuk, V. M. and Suleimanova, G. S.: Normal structure of the adjoint group in radical rings Rn(K,J), Sibirsk. Mat. Zh. 43(2) (2002), 519–537.
Levchuk, V. M. and Voitenko, T. Yu.: Zenkov’s hypothesis about paired p-Sylow intersections of Chevalley groups by characteristics p, Ukr. Mat. Zh. 54(7) (2002), 907–918.
Merzlyakov, Yu. I.: Equisubgroups of unitriangular groups: A criterion of self normalizability, Dokl. Russ. Akad. Nauk 50(3) (1995), 507–511.
Suleimanova, G. S.: A normal structure of unipotent subgroup of the unitary group over a field, In: Proceed. 2-nd Intern. Conf. “Symmetry and Diff. Equations”, Inst. Comput. Modelling, Krasnoyarsk, 2000, pp. 206–209.
Suzuki, M.: Finite groups even order in which Sylow 2-groups are independent, Ann. Math. 80 (1964), 58–77.
Voitenko, T. Yu.: Pair unipotent intersections of Chevalley groups of Lie types El, In: Proceed. of Krasnoyarsk State University, KSU, Krasnoyarsk, 2001, pp. 32–40.
Zenkov, V. I.: Structure of intersections of nilpotent π-subgroups in π-soluble finite groups, Sibirsk. Mat. Zh. 34(4) (1993), 103–115.
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This research is supported by Russian fund of fundamental researches (RFFI), grant 03-01-00905.
Mathematics Subject Classifications (2000)
13C13, 08A35, 20F28, 20D06, 20D20, 20F406.
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Levchuk, V.M. Sylow Subgroups of the Chevalley Groups and Associated (Weakly) Finitary Groups and Rings. Acta Appl Math 85, 225–232 (2005). https://doi.org/10.1007/s10440-004-5624-2
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DOI: https://doi.org/10.1007/s10440-004-5624-2