Geodesic
Automorphisms of Chevalley groups of type $G_2$ over local rings without~$1/2$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 7, pp. 49-66.

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In this paper, we prove that every automorphism of a Chevalley group of type $G_2$ over a commutative local ring without $1/2$ is the composition of a ring automorphism and a conjugation by some matrix. 
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E. I. Bunina; P. A. Veryovkin. Automorphisms of Chevalley groups of type $G_2$ over local rings without~$1/2$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 7, pp. 49-66. http://geodesic.mathdoc.fr/item/FPM_2012_17_7_a3/

[1] Bunina E. I., “Avtomorfizmy grupp Shevalle tipov $B_2$ i $G_2$ nad lokalnymi koltsami”, Fundament. i prikl. mat., 13:4 (2007), 3–29 | MR | Zbl

[2] Bunina E. I., “Avtomorfizmy grupp Shevalle tipa $B_l$ nad lokalnymi koltsami s $1/2$”, Fundament. i prikl. mat., 15:7 (2009), 3–46 ; arXiv: 0911.4243 | MR

[3] Bunina E. I., “Avtomorfizmy grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s neobratimoi dvoikoi”, Fundament. i prikl. mat., 15:7 (2009), 47–80 | MR

[4] Bunina E. I., “Avtomorfizmy grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s 1/2”, Fundament. i prikl. mat., 15:2 (2009), 35–59 ; arXiv: 0907.5595 | MR

[5] Bunina E. I., “Avtomorfizmy elementarnykh prisoedinennykh grupp Shevalle tipov $A_l$, $D_l$, $E_l$ nad lokalnymi koltsami s 1/2”, Algebra i logika, 48:4 (2009), 443–470 ; arXiv: math/0702046 | MR | Zbl

[6] Petechuk V. M., “Avtomorfizmy grupp $\mathrm{SL}_n$, $\mathrm{GL}_n$ nad nekotorymi lokalnymi koltsami”, Mat. zametki, 28:2 (1980), 187–204 | MR | Zbl

[7] Petechuk V. M., “Avtomorfizmy grupp $\mathrm{SL}_3(K)$, $\mathrm{GL}_3(K)$”, Mat. zametki, 31:5 (1982), 657–668 | MR | Zbl

[8] Petechuk V. M., “Avtomorfizmy matrichnykh grupp nad kommutativnymi koltsami”, Mat. sb., 117(159):4 (1982), 534–547 | MR | Zbl

[9] Shevalle K., “O nekotorykh prostykh gruppakh”, Matematika: Period. sb. perev. inostr. statei, 2:1 (1958), 3–58 | MR

[10] Abe E., “Chevalley groups over local rings”, Tôhoku Math. J., 21:3 (1969), 474–494 | DOI | MR | Zbl

[11] Eiichi Abe, “Avtomorfizmy grupp Shevalle nad kommutativnymi koltsami”, Algebra i analiz, 5:2 (1993), 74–90 | MR | Zbl

[12] Borel A., Tits J., “Homomorphismes “abstraits” de groupes algébriques simples”, Ann. Math., 73 (1973), 499–571 | DOI | MR

[13] Bourbaki N., Groupes et algébres de Lie, Hermann, 1968 | MR

[14] Bunina E. I., “Automorphisms of Chevalley groups of type $F_4$ over local rings with $1/2$”, J. Algebra, 323 (2010), 2270–2289 ; arXiv: 0907.5592 | DOI | MR | Zbl

[15] Bunina E. I., “Automorphisms of Chevalley groups of different types over commutative rings”, J. Algebra, 355:1 (2012), 154–170 | DOI | MR | Zbl

[16] Carter R. W., Simple Groups of Lie Type, Wiley, London, 1989 | MR | Zbl

[17] Carter R. W., Chen Yu., “Automorphisms of affine Kac–Moody groups and related Chevalley groups over rings”, J. Algebra, 155 (1993), 44–94 | DOI | MR | Zbl

[18] Chen Yu., “Isomorphic Chevalley groups over integral domains”, Rend. Sem. Mat. Univ. Padova, 92 (1994), 231–237 | MR | Zbl

[19] Chen Yu., “Automorphisms of simple Chevalley groups over $\mathbb Q$-algebras”, Tôhoku Math. J., 348 (1995), 81–97 | DOI | MR

[20] Chen Yu., “On representations of elementary subgroups of Chevalley groups over algebras”, Proc. Amer. Math. Soc., 123:8 (1995), 2357–2361 | DOI | MR | Zbl

[21] Chen Yu., “Isomorphisms of adjoint Chevalley groups over integral domains”, Trans. Amer. Math. Soc., 348:2 (1996), 1–19 | DOI | MR

[22] Chen Yu., “Isomorphisms of Chevalley groups over algebras”, J. Algebra, 226 (2000), 719–741 | DOI | MR | Zbl

[23] Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer, New York, 1978 | MR | Zbl

[24] Humphreys J. F., “On the automorphisms of infinite Chevalley groups”, Can. J. Math., 21 (1969), 908–911 | DOI | MR | Zbl

[25] Klyachko A. A., Automorphisms and isomorphisms of Chevalley groups and algebras, 2007, arXiv: 0708.2256v3 | MR

[26] McDonald B. R., “Automorphisms of $\mathrm{GL}_n(R)$”, Trans. Amer. Math. Soc., 215 (1976), 145–159 | MR | Zbl

[27] Steinberg R., “Automorphisms of finite linear groups”, Can. J. Math., 121 (1960), 606–615 | DOI | MR

[28] Steinberg R., Lectures on Chevalley Groups, Yale Univ., 1967 | MR

[29] Vavilov N. A., Plotkin E. B., “Chevalley groups over commutative rings. I. Elementary calculations”, Acta Appl. Math., 45 (1996), 73–115 | DOI | MR