Skip to main content
SpringerLink
Log in

On the class of groups with pronormal hall π-subgroups

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

Given a set π of prime numbers, we define the class of all finite groups in which Hall π-subgroups exist and are pronormal by analogy with the Hall classes , , and . We study whether is closed under the main class-theoretic closure operations. In particular, we establish that is a saturated formation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hall P., “Theorems like Sylow’s,” Proc. London Math. Soc., 6, No. 22, 286–304 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  2. Vdovin E. P., “Carter subgroups of finite almost simple groups,” Algebra and Logic, 46, No. 2, 90–119 (2007).

    Article  MathSciNet  Google Scholar 

  3. Vdovin E. P. and Revin D. O., “Theorems of Sylow type,” Russian Math. Surveys, 66, No. 5, 829–870 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  4. Vdovin E. P. and Revin D. O., “On the pronormality of Hall subgroups,” Siberian Math. J., 54, No. 1, 22–28 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  5. Vdovin E. P. and Revin D. O., “Pronormality of Hall subgroups in finite simple groups,” Siberian Math. J., 53, No. 3, 419–430 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  6. Revin D. O., “Around a conjecture of P. Hall,” Siberian Electronic Math. Reports, 6, 366–380 (2009).

    MATH  MathSciNet  Google Scholar 

  7. Gross F., “Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., 19, No. 4, 311–319 (1987).

    Article  MATH  MathSciNet  Google Scholar 

  8. Guo W., The Theory of Classes of Groups, Kluwer Acad. Publ., Beijing and New York (2006).

    Google Scholar 

  9. Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).

    Book  MATH  Google Scholar 

  10. Revin D. O. and Vdovin E. P., “Generalizations of the Sylow theorem,” in: Groups St. Andrews, 2009 in Bath. V. 2 (eds. C. M. Campbell, M. R. Quick, E. F. Robertson, C. M. Roney-Dougal, G. C. Smith, G. Traustason), Cambridge Univ. Press, Cambridge, 2011, pp. 488–519 (London Math. Soc. Lect. Notes Ser.; V. 388).

    Chapter  Google Scholar 

  11. Chunikhin S. A., Subgroups of Finite Groups, Wolters-Noordhoff, Groningen (1969).

    MATH  Google Scholar 

  12. Wielandt H., “Entwicklungslinien in der Strukturtheorie der endlichen Gruppen,” in: Proc. Int. Congress Math. (Edinburgh, 1958), Cambridge Univ. Press, London, 1960, pp. 268–278.

    Google Scholar 

  13. Suzuki M., Group Theory. II, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo (1986).

    Book  MATH  Google Scholar 

  14. Shemetkov L. A., Formations of Finite Groups [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  15. Mazurov V. D. and Khukhro E. I. (Eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 17th ed., Sobolev Institute of Mathematics, Novosibirsk (2010).

    Google Scholar 

  16. Vdovin E. P., Revin D. O., and Shemetkov L. A., “Formations of finite Cπ-groups,” St. Petersburg Math. J., 24, No. 1, 29–37 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  17. Vdovin E. P. and Revin D. O., “A conjugacy criterion for Hall subgroups in finite groups,” Siberian Math. J., 51, No. 3, 402–409 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  18. Revin D. O. and Vdovin E. P., “An existence criterion for Hall subgroups of finite groups,” J. Group Theory, 14, No. 1, 93–101 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  19. Chunikhin S. A., “On Sylow-regular groups,” Dokl. Akad. Nauk SSSR, 60, No. 5, 773–774 (1948).

    MATH  Google Scholar 

  20. Chunikhin S. A., “On the existence and conjugacy of subgroups of a finite group,” Mat. Sb., 33, No. 1, 111–132 (1953).

    Google Scholar 

  21. Vdovin E. P. and Revin D. O., “The nonradicality of the class Eπ-groups,” Proc. Inst. Math. NAS Belarus, 20, No. 1, 35–39 (2013).

    MathSciNet  Google Scholar 

  22. Revin D. O. and Vdovin E. P., “Hall subgroups of finite groups,” Contemp. Math., 402, 229–265 (2006).

    Article  MathSciNet  Google Scholar 

  23. Gross F., “On the existence of Hall subgroups,” J. Algebra, 98, No. 1, 1–13 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  24. Guo W. and Li B., “On the Shemetkov problem for Fitting classes,” Beiträge Algebra Geom., 48, No. 1, 281–289 (2007).

    MATH  Google Scholar 

  25. Guo W., “Formations determined by Hall subgroups,” J. Appl. Algebra Discrete Struct., 4, No. 3, 139–147 (2006).

    MATH  MathSciNet  Google Scholar 

  26. Guo W., “Some problems and results for the research on Sylow objects of finite groups,” J. Xuzhou Norm. Univ. Nat. Sci. Ed., 23, No. 3, 1–6 (2005).

    MATH  MathSciNet  Google Scholar 

  27. Hartley B., “A theorem of Sylow type for a finite groups,” Math. Z., Bd 122, Heft 4, 223–226 (1971).

    Google Scholar 

  28. Revin D. O. and Vdovin E. P., “On the number of classes of conjugate Hall subgroups in finite simple groups,” J. Algebra, 324, No. 12, 3614–3652 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  29. Vdovin E. P., Manzaeva N. Ch., and Revin D. O., “On the heritability of the property Dπ by subgroups,” Proc. Steklov Inst. Math., Suppl. 1, 279, S130–S138 (2012).

    Article  Google Scholar 

  30. Vdovin E. P. and Revin D. O., “Hall subgroups of odd order in finite groups,” Algebra and Logic, 41, No. 1, 8–29 (2002).

    Article  MathSciNet  Google Scholar 

  31. Vorob’ev S. N. and Zalesskaya E. N., “An analog of Shemetkov’s conjecture for Fischer classes of finite groups,” Siberian Math. J., 54, No. 5, 790–797 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  32. Zagurskiĭ V. N. and Vorob’ev N. T., “Fitting classes with given properties of Hall subgroups,” Math. Notes, 78, No. 2, 213–218 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  33. Mazurov V. D. and Revin D. O., “On the Hall Dπ-property for finite groups,” Siberian. Math. J., 38, No. 1, 106–113 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  34. Revin D. O., “The Dπ-property in a class of finite groups,” Algebra and Logic, 41, No. 3, 187–206 (2002).

    Article  MathSciNet  Google Scholar 

  35. Revin D. O., “The Dπ-property in finite simple groups,” Algebra and Logic, 47, No. 3, 210–227 (2008).

    Article  MathSciNet  Google Scholar 

  36. Yi Xiaolan and Shemetkov L. A., “The formation of finite groups with a supersolvable π-Hall subgroup,” Math. Notes, 87, No. 2, 258–263 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  37. Chunikhin S. A., “On Sylow properties of finite groups,” Dokl. Akad. Nauk SSSR, 73, No. 1, 29–32 (1950).

    MATH  Google Scholar 

  38. Shemetkov L. A., “A new D-theorem in the theory of finite groups,” Soviet Math. Dokl., 6, No. 2, 89–93 (1965).

    MATH  Google Scholar 

  39. Shemetkov L. A., “Sylow properties of finite groups,” Math. USSR-Sb., 5, No. 2, 261–274 (1968).

    Article  MATH  Google Scholar 

  40. Shemetkov L. A., “On Sylow properties of finite groups,” Dokl. Akad. Nauk BSSR, 16, No. 10, 881–883 (1972).

    MATH  MathSciNet  Google Scholar 

  41. Shemetkov L. A., “Two directions in the development of the theory of non-simple finite groups,” Russian Math. Surveys, 30, No. 2, 185–206 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  42. Shemetkov L. A., “Generalizations of Sylow’s theorem,” Siberian Math. J., 44, No. 6, 1127–1132 (2003).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, P. R. China

    W. Guo

  2. Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, Russia

    D. O. Revin

Authors
  1. W. Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. O. Revin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to W. Guo.

Additional information

Original Russian Text Copyright © 2014 Guo W. and Revin D.O.

The first author was supported by the NNSF of China (Grant 11371335) and the Chinese Universities Scientific Fund (Project WK0010000029). The second author was supported by the Russian Foundation for Basic Research (Grant 13-01-00505) and the Target Program of the Siberian Division of the Russian Academy of Sciences for 2012–2014 (Integration Project No. 14).

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 3, pp. 509–524, May–June, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, W., Revin, D.O. On the class of groups with pronormal hall π-subgroups. Sib Math J 55, 415–427 (2014). https://doi.org/10.1134/S0037446614030033

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446614030033

Keywords

Search

Navigation