Geodesic
Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 133-142
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The description of the automorphisms of an unipotent subgroup $U$ of a Chevalley group over a field $K$ known earlier for $\operatorname{char}K\ne2,3$ (Gibbs, 1970) was completed in 1990 together with a solution of problem (1.5) from A. S. Kondrat’ev's survey (Usp. Mat. Nauk, 1986). In the present paper, $\operatorname{Aut}U$ is described for the case of finitary Chevalley groups. For a Chevalley group of classical type, it is proved that any large Abelian subgroup from $U$ is conjugate to a normal subgroup in $U$. It is shown that this is not so in the general case; therefore, problem (1.6) from Kondrat'ev's survey about large Abelian subgroups in $U$ is reduced to listing the exceptions. Large Abelian normal subgroups were listed by the authors earlier.
Keywords: finitary Chevalley group, unipotent subgroup, large abelian subgroup.
Mots-clés : automorphism 
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V. M. Levchuk; G. S. Suleimanova. Automorphisms and normal structure of unipotent subgroups of finitary Chevalley groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 2, pp. 133-142. http://geodesic.mathdoc.fr/item/TIMM_2009_15_2_a12/

[1] Vdovin E. P., “Maksimalnye poryadki abelevykh podgrupp v konechnykh gruppakh Shevalle”, Mat. zametki, 69:4 (2001), 524–549 | MR | Zbl

[2] Vdovin E. P., “Bolshie abelevy unipotentnye podgruppy konechnykh grupp Shevalle”, Algebra i logika, 40:5 (2001), 523–544 | MR | Zbl

[3] Gorchakov Yu. M., Gruppy s konechnymi klassami sopryazhennykh elementov, Nauka, M., 1978, 119 pp. | MR | Zbl

[4] Kondratev A. S., “Podgruppy konechnykh grupp Shevalle”, Uspekhi mat. nauk, 41:1(247) (1986), 57–96 | MR

[5] Levchuk V. M., “Avtomorfizmy unipotentnykh podgrupp grupp lieva tipa malykh rangov”, Algebra i logika, 29:2 (1990), 141–161 | MR | Zbl

[6] Levchuk V. M., “Avtomorfizmy unipotentnykh podgrupp grupp Shevalle”, Algebra i logika, 29:3 (1990), 315–338 | MR | Zbl

[7] Levchuk V. M., “Nekotorye lokalno nilpotentnye koltsa i ikh prisoedinennye gruppy”, Mat. zametki, 42:5 (1987), 631–641 | MR | Zbl

[8] Levchuk V. M., Suleimanova G. S., “Normalnoe stroenie unipotentnoi podgruppy gruppy lieva tipa i smezhnye voprosy”, Dokl. RAN, 419:5 (2008), 595–598 | MR | Zbl

[9] Maltsev A. I., “Kommutativnye podalgebry poluprostykh algebr Li”, Izv. AN SSSR. Ser. mat., 9:4 (1945), 291–300

[10] Merzlyakov Yu. I., “Ekvipodgruppy unitreugolnykh grupp: kriterii samonormalizuemosti”, Dokl. RAN, 339:6 (1994), 732–735 | Zbl

[11] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975, 264 pp. | MR | Zbl

[12] Barry M. J. J., “Large Abelian subgroups of Chevalley groups”, J. Austral. Math. Soc. Ser. A, 27:1 (1979), 59–87 | DOI | MR | Zbl

[13] Barry M. J. J., Wong W. J., “Abelian 2-subgroups of finite symplectic groups in characteristic 2”, J. Austral. Math. Soc. Ser. A, 33:3 (1982), 345–350 | DOI | MR | Zbl

[14] Carter R., Simple groups of Lie type, Wiley and Sons, New York, 1972, 332 pp. | MR | Zbl

[15] Gibbs J., “Automorphisms of certain unipotent groups”, J. Algebra, 14:2 (1970), 203–228 | DOI | MR | Zbl

[16] Gupta Ch. K., Levchuk V. M. and Ushakov Yu. Yu., “Hypercentral and monic automorphisms of classical algebras, rings and groups”, J. Sib. Fed. Univ. Math. Phys., 1:4 (2008), 380–390

[17] Kuzucuoglu F., Levchuk V. M., “Isomorphism of certain locally nilpotent finitary groups and associated rings”, Acta Appl. Math., 82:2 (2004), 169–181 | DOI | MR | Zbl

[18] Levchuk V. M., “Chevalley groups and their unipotent subgroups”, Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989), Contemp. Math., 131, Amer. Math. Soc., Providence, RI, 1992, 227–242 | MR

[19] Levchuk V. M., “Sylow subgroups of the Chevalley groups and associated (weakly) finitary groups and rings”, Acta Appl. Math., 85:1–3 (2005), 225–232 | DOI | MR | Zbl

[20] Weir A. J., “Sylow $p$-subgroups of the general linear group over finite fields of characteristic $p$”, Proc. Amer. Math. Soc., 6:3 (1955), 454–464 | DOI | MR | Zbl

[21] Wong W. J., “Abelian unipotent subgroups of finite orthogonal groups”, J. Austral. Math. Soc. Ser. A, 32:2 (1982), 223–245 | DOI | MR | Zbl

[22] Wong W. J., “Abelian unipotent subgroups of finite unitary and symplectic groups”, J. Austral. Math. Soc. Ser. A, 33:2 (1982), 331–344 | DOI | MR | Zbl