On Characterization of Simple Orthogonal Groups of Odd Dimension in the Class of Periodic Groups


Suppose that \( n \) is an integer, \( n\geq 3 \). We prove that a periodic group saturated with a set of the finite simple groups \( O_{2n+1}(q) \), where \( q \) is congruent to \( \pm 3 \) modulo 8, is isomorphic to \( O_{2n+1}(F) \) for some locally finite field \( F \).

This is a preview of subscription content, access via your institution.


  1. 1.

    Carter R. W., Simple Groups of Lie Type, John Wiley and Sons, London (1972).

    Google Scholar 

  2. 2.

    Belyaev V. V., “Locally finite Chevalley groups,” in: Studies in Group Theory [Russian], Ural Scientific Center, Sverdlovsk (1984), 39–50.

  3. 3.

    Borovik A. V., “Embeddings of finite Chevalley groups and periodic linear groups,” Sib. Math. J., vol. 24, no. 6, 843–851 (1983).

    Article  Google Scholar 

  4. 4.

    Hartley B. and Shute G., “Monomorphisms and direct limits of finite groups of Lie type,” Q. J. Math. Oxford, Ser. 2, vol. 35, no. 137, 49–71 (1984).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Thomas S., “The classification of the simple periodic linear groups,” Arch. Math., vol. 41, no. 2, 103–116 (1983).

    MathSciNet  Article  Google Scholar 

  6. 6.

    Larsen M. J. and Pink R., “Finite subgroups of algebraic groups,” J. Amer. Math. Soc., vol. 24, no. 4, 1105–1158 (2011).

    MathSciNet  Article  Google Scholar 

  7. 7.

    Shlepkin A. K., “On some periodic groups saturated by finite simple groups,” Siberian Adv. Math., vol. 9, no. 2, 100–108 (1999).

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Rubashkin A. G. and Filippov K. A., “Periodic groups saturated with the groups \( L_{2}(p^{n}) \),” Sib. Math. J., vol. 46, no. 6, 1119–1122 (2005).

    Article  Google Scholar 

  9. 9.

    Lytkina D. V. and Shlepkin A. K., “Periodic groups saturated with finite simple groups of types \( U_{3} \) and \( L_{3} \),” Algebra and Logic, vol. 55, no. 4, 289–294 (2016).

    MathSciNet  Article  Google Scholar 

  10. 10.

    Filippov K. A., Groups Saturated with Finite Nonabelian Groups and Their Extensions [Russian]. Cand. Sci. Math. Dissertation, Siberian Federal University, Krasnoyarsk (2005).

    Google Scholar 

  11. 11.

    Filippov K. A., “On periodic groups saturated by finite simple groups,” Sib. Math. J., vol. 53, no. 2, 345–351 (2012).

    MathSciNet  Article  Google Scholar 

  12. 12.

    Lytkina D. V. and Mazurov V. D., “Characterization of simple symplectic groups of degree 4 over locally finite fields in the class of periodic groups,” Algebra and Logic, vol. 57, no. 3, 201–210 (2018).

    MathSciNet  Article  Google Scholar 

  13. 13.

    Wei X., Guo W., Lytkina D. V., and Mazurov V. D., “Characterization of locally finite simple groups of the type \( {}^{3}D_{4} \) over fields of odd characteristic in the class of periodic groups,” Sib. Math. J., vol. 59, no. 5, 799–804 (2018).

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lytkina D. V. and Mazurov V. D., “On the periodic groups saturated with finite simple groups of Lie type \( B_{3} \),” Sib. Math. J., vol. 61, no. 3, 499–503 (2020).

    Article  Google Scholar 

  15. 15.

    Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Clarendon, Oxford (1985).

    Google Scholar 

  16. 16.

    Taylor D. E., The Geometry of the Classical Groups, Heldermann, Berlin (1992).

    Google Scholar 

  17. 17.

    Kleidman P. B. and Liebeck M., The Subgroup Structure of the Finite Classical Groups, Cambridge Univ., Cambridge (1990).

    Google Scholar 

  18. 18.

    Bray J. N., Holt D. F., and Roney-Dougal C. M., The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, Cambridge Univ., Cambridge (2013) (Lond. Math. Soc. Lect. Note Ser.).

    Google Scholar 

  19. 19.

    Dickson L. E., “Representation of the general symmetric group as linear groups in finite and infinite fields,” Trans. Amer. Math. Soc., vol. 9, no. 2, 121–148 (1908).

    MathSciNet  Article  Google Scholar 

  20. 20.

    Wagner A., “The faithful linear representations of least degree of \( S_{n} \) and \( A_{n} \) over field of characteristic 2,” Math. Z., vol. 151, no. 2, 127–137 (1976).

    MathSciNet  Article  Google Scholar 

  21. 21.

    Wong W. J., “Twisted wreath products and Sylow 2-subgroups of classical simple groups,” Math. Z., vol. 97, no. 5, 406–424 (1967).

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kondrat’ev A. S., “Normalizers of the Sylow 2-subgroups in finite simple groups,” Math. Notes, vol. 78, no. 3, 338–346 (2005).

    MathSciNet  Article  Google Scholar 

  23. 23.

    Lytkina D. V., Tukhvatullina L. R., and Filippov K. A., “The periodic groups saturated by finitely many finite simple groups,” Sib. Math. J., vol. 49, no. 2, 317–321 (2008).

    MathSciNet  Article  Google Scholar 

Download references


The work of D. V. Lytkina was supported by the Mathematical Center in Akademgorodok under Agreement No. 075–15–2019–1613 with the Ministry of Science and Higher Education of the Russian Federation; the work of V. D. Mazurov was supported by the Russian Science Foundation (Project 19–11–00039).

Author information



Corresponding author

Correspondence to D. V. Lytkina.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lytkina, D.V., Mazurov, V.D. On Characterization of Simple Orthogonal Groups of Odd Dimension in the Class of Periodic Groups. Sib Math J 62, 77–83 (2021). https://doi.org/10.1134/S0037446621010080

Download citation


  • periodic group
  • group saturated with a set of groups
  • locally finite group


  • 512.542