Skip to main content
Log in

Automorphisms of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. E. Abe, “Chevalley groups over local rings,” Tôhoku Math. J., 21, No. 3, 474–494 (1969).

    Article  MATH  Google Scholar 

  2. E. Abe, “Automorphisms of Chevalley groups over commutative rings,” Algebra Anal., 5, No. 2, 74–90 (1993).

    Google Scholar 

  3. A. Borel and J. Tits, “Homomorphismes “abstraits” de groupes algébriques simples,” Ann. Math., 73, 499–571 (1973).

    Article  MathSciNet  Google Scholar 

  4. N. Bourbaki, Groupes et algébres de Lie, Hermann (1968).

  5. E. I. Bunina, “Automorphisms of adjoint Chevalley groups of types B 2 and G 2 over local rings,” J. Math. Sci., 155, No. 6, 795–814 (2008).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. I. Bunina, “Automorphisms and normalizers of Chevalley groups of types A l , D l , E l over local rings with 1/2,” Fundam. Prikl. Mat., 15, No. 2, 35–59, arXiv:0907.5595 (2009).

  7. E. I. Bunina, “Automorphisms of elementary adjoint Chevalley groups of types A l , D l , E l over local rings,” Algebra Logic, 48, No. 4, 250–267, arXiv:math/0702046 (2009).

  8. E. I. Bunina, “Automorphisms of Chevalley groups of types A l , D l , E l over local rings without 1/2,” Fundam. Prikl. Mat., 15, No. 7, 47–80 (2009).

    MathSciNet  Google Scholar 

  9. E. I. Bunina, “Automorphisms of Chevalley groups of types B l over local rings with 1/2,” Fundam. Prikl. Mat., 15, No. 7, 3–46, arXiv:0911.4243 (2009).

  10. E. I. Bunina, “Automorphisms of Chevalley groups of type F 4 over local rings with 1/2,” J. Algebra, 323, 2270–2289, arXiv:0907.5592 (2010).

    Google Scholar 

  11. E. I. Bunina, “Automorphisms of Chevalley groups of different types over commutative rings,” J. Algebra, 355, No. 1, 154–170 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  12. R. W. Carter, Simple Groups of Lie Type, Wiley, London (1989).

    MATH  Google Scholar 

  13. R. W. Carter and Yu Chen, “Automorphisms of affine Kac–Moody groups and related Chevalley groups over rings,” J. Algebra, 155, 44–94 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  14. Yu Chen, “Isomorphic Chevalley groups over integral domains,” Rend. Sem. Mat. Univ. Padova, 92, 231–237 (1994).

  15. Yu Chen, “Automorphisms of simple Chevalley groups over \( \mathbb{Q} \)-algebras,” Tôhoku Math. J., 348, 81–97 (1995).

    Google Scholar 

  16. Yu Chen, “On representations of elementary subgroups of Chevalley groups over algebras.” Proc. Am. Math. Soc., 123, No. 8, 2357–2361 (1995).

    Article  MATH  Google Scholar 

  17. Yu Chen, “Isomorphisms of adjoint Chevalley groups over integral domains,” Trans. Am. Math. Soc., 348, No. 2, 1–19 (1996).

    Article  Google Scholar 

  18. Yu Chen, “Isomorphisms of Chevalley groups over algebras,” J. Algebra, 226, 719–741 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Chevalley, “Certain schemas des groupes semi-simples,” Sem. Bourbaki, 219, 1–16 (1960–1961).

    Google Scholar 

  20. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York (1978).

    MATH  Google Scholar 

  21. J. F. Humphreys, “On the automorphisms of infinite Chevalley groups,” Can. J. Math., 21, 908–911 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  22. A. A. Klyachko, Automorphisms and Isomorphisms of Chevalley Groups and Algebras, arXiv:math/0708.2256v3 (2007).

  23. B. R. McDonald, “Automorphisms of GL n (R),” Trans. Am. Math. Soc., 215, 145–159 (1976).

    MATH  Google Scholar 

  24. V. M. Petechuk, “Automorphisms of groups SL n , GL n over some local rings,” Mat. Zametki, 28, No. 2, 187–206 (1980).

    MATH  MathSciNet  Google Scholar 

  25. V. M. Petechuk, “Automorphisms of groups SL3(K), GL3(K),” Mat. Zametki, 31, No. 5, 657–668 (1982).

    MATH  MathSciNet  Google Scholar 

  26. V. M. Petechuk, “Automorphisms of matrix groups over commutative rings,” Mat. Sb., 45, 527–542 (1983).

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  28. R. Steinberg, Lectures on Chevalley Groups, Yale Univ. (1967).

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to E. I. Bunina.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 7, pp. 49–66, 2011/12.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bunina, E.I., Veryovkin, P.A. Automorphisms of Chevalley Groups of Type G 2 Over Local Rings Without 1/2. J Math Sci 197, 479–491 (2014). https://doi.org/10.1007/s10958-014-1729-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-014-1729-y

Keywords

Navigation