Skip to main content
SpringerLink
Log in

The Bogomolov multiplier of rigid finite groups

  • Published:
Archiv der Mathematik Aims and scope Submit manuscript

Abstract

The Bogomolov multiplier of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. This invariant of G plays an important role in birational geometry of quotient spaces V/G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.

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. V. Bardakov, A. Vesnin, and M. K. Yadav, Class preserving automorphisms of unitriangular groups, Internat. J. Algebra Computation 22 (2012), 12500233, 17 pp.

    Google Scholar 

  2. Ya. Berkovich, Groups of prime power order, Vol. 1, Walter de Gruyter, Berlin, 2008.

  3. Blackburn N.: , Finite groups in which the nonnormal subgroups have nontrivial intersection, J. Algebra 3, 30–37 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  4. F. A. Bogomolov, The Brauer group of quotient spaces by linear group actions, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 485–516;

    Google Scholar 

  5. Bogomolov F. A.: The Brauer group of quotient spaces by linear group actions, English transl. in Math. USSR Izv. 30, 455–485 (1988)

    Article  MATH  Google Scholar 

  6. F. Bogomolov and C. Böhning, Isoclinism and stable cohomology of wreath products, in: “Birational Geometry, Rational Curves, and Arithmetic” (F. Bogomolov, B. Hassett, Yu. Tschinkel, Eds.), Springer, New York, 2013, pp. 57–76.

  7. F. Bogomolov, J. Maciel, and T. Petrov, Unramified Brauer groups of finite simple groups of Lie type A ℓ , Amer. J. Math. 126 (2004), 935–949.

  8. W. Burnside, Theory of groups of finite order, 2nd ed., Cambridge Univ. Press, Cambridge, 1911; reprinted by Dover Publications, New York, 1955.

  9. Burnside W.: On the outer automorphisms of a group, Proc. London Math. Soc. 11, 40–42 (1913)

    Article  Google Scholar 

  10. J. F. Carlson and J. Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000), 303–335.

    Google Scholar 

  11. H. Chu, S.-J. Hu, M. Kang, and B. E. Kunyavskiǐ, Noether’s problem and the unramified Brauer group for groups of order 64, Intern. Math. Research Notices 2010, no. 12, 2329–2366.

  12. H. Chu, S.-J. Hu, M. Kang, and Yu. G. Prokhorov, Noether’s problem for groups of order 32, J. Algebra 320 (2008), 3022–3035.

    Google Scholar 

  13. Chu H., Kang M.: Rationality of p-group actions, J. Algebra 237, 673–690 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feit W., Seitz G. M.: On finite rational groups and related topics, Illinois J. Math. 33, 103–131 (1989)

    MATH  Google Scholar 

  15. M. Fuma and Y. Ninomiya, “Hasse principle” for finite p-groups with cyclic subgroups of index p 2, Math. J. Okayama Univ. 46 (2004), 31–38.

    Google Scholar 

  16. D. Gorenstein, Finite groups, Harper and Row, N. Y., 1968.

  17. Herman A., Li Y.: Class preserving automorphisms of Blackburn groups, J. Australian Math. Soc. 80, 351–358 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Hertweck, Class-preserving automorphisms of finite groups, J. Algebra 241 (2001), 1–26.

    Google Scholar 

  19. M. Hertweck, Contributions to the integral representation theory of groups, Habilitationsschrift, Univ. Stuttgart, 2004, available online at http://elib.unistuttgart.de/opus/volltexte/2004/1638.

  20. M. Hertweck and E. Jespers, Class-preserving automorphisms and the normalizer property for Blackburn groups, J. Group Theory 12 (2009), 157–169.

    Google Scholar 

  21. Higgs R. J.: Subgroups of the Schur multiplier, J. Austral. Math. Soc. Ser. A 48, 497–505 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Hoshi, M. Kang, and B. E. Kunyavskii, Noether’s problem and unramified Brauer groups, Asian J. Math. 17 (2013), 689–714.

    Google Scholar 

  23. S.-J. Hu and M. Kang, Noether’s problem for some p-groups, in: “Cohomological and geometric approaches to rationality problems”, Progr. Math., vol. 282, Birkhäuser, Boston, MA, 2010, pp. 149–162.

  24. M. Kang, Noether’s problem for p-groups with a cyclic subgroup of index p 2, Adv. Math. 226 (2011), 218–234.

    Google Scholar 

  25. Kang M.: Retract rational fields, J. Algebra 349, 22–37 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kang M.: Bogomolov multipliers and retract rationality for semi-direct products, J. Algebra 397, 407–425 (2014)

    Article  MathSciNet  Google Scholar 

  27. G. Karpilovsky, Group Representations, vol. 2, North-Holland Math. Stud. 177, North-Holland, Amsterdam, 1993.

  28. M. Kumar and L. R. Vermani, “Hasse principle” for extraspecial p-groups, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), 123–125

    Google Scholar 

  29. M. Kumar and L. R. Vermani, “Hasse principle” for groups of order p 4, Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), 95–98.

  30. M. Kumar and L. R. Vermani, On automorphisms of some p-groups, Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), 46–50.

    Google Scholar 

  31. B. E. Kunyavskiĭ, The Bogomolov multiplier of finite simple groups, in: “Cohomological and geometric approaches to rationality problems”, Progr. Math., vol. 282, Birkhäuser, Boston, MA, 2010, pp. 209–217.

  32. B. Kunyavskiĭ, Local-global invariants of finite and infinite groups: around Burnside from another side, Expos. Math. 31 (2013), 256–273.

    Google Scholar 

  33. Laue R.: On outer automorphism groups, Math. Z. 148, 177–188 (1976)

    MATH  MathSciNet  Google Scholar 

  34. I. M. Michailov, Bogomolov multipliers for some p-groups of nilpotency class 2, arXiv:1307.0738.

  35. I. M. Michailov, Bogomolov multipliers for unitriangular groups, arXiv:1308.3408.

  36. Moravec P.: Groups of order p 5 and their unramified Brauer groups, J. Algebra 372, 320–327 (2012)

    MathSciNet  Google Scholar 

  37. Moravec P.: Unramified Brauer groups and isoclinism, Ars Math. Contemp. 7, 337–340 (2014)

    Google Scholar 

  38. Ono T.: A note on Shafarevich–Tate sets for finite groups, Proc. Japan Acad. Ser. A Math. Sci. 74, 77–79 (1998)

    Article  MATH  Google Scholar 

  39. T. Ono and H. Wada, “Hasse principle” for symmetric and alternating groups, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999), 61–62.

  40. Peyre E.: Unramified cohomology of degree 3 and Noether’s problem, Invent. Math. 171, 191–225 (2008)

    MATH  MathSciNet  Google Scholar 

  41. Saltman D. J.: Noether’s problem over an algebraically closed field, Invent. Math. 77, 71–84 (1984)

    Google Scholar 

  42. Saltman D. J.: The Brauer group and the center of generic matrices, J. Algebra 97, 53–67 (1985)

    Article  MathSciNet  Google Scholar 

  43. Suzuki M.: On finite groups with cyclic Sylow subgroups for all odd primes, Amer. J. Math. 77, 657–691 (1955)

    Article  MATH  Google Scholar 

  44. C. T. C. Wall, On the structure of finite groups with periodic cohomology, preprint, 2010, available at http://www.liv.ac.uk/~ctcw/FGPCnew.

  45. Yadav M. K.: On automorphisms of some finite p-groups, Proc. Indian Acad. Sci. (Math. Sci.) 118, 1–11 (2008)

    Google Scholar 

  46. M. K. Yadav, Class preserving automorphisms of finite p-groups: A survey, in: “Groups St. Andrews 2009 in Bath., vol. 2”, London Math. Soc. Lecture Note Ser., vol. 388, Cambridge Univ. Press, Cambridge et al., 2011, pp. 569– 579.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan

    Ming-chang Kang

  2. Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel

    Boris Kunyavskiĭ

Authors
  1. Ming-chang Kang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris Kunyavskiĭ
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris Kunyavskiĭ.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kang, Mc., Kunyavskiĭ, B. The Bogomolov multiplier of rigid finite groups. Arch. Math. 102, 209–218 (2014). https://doi.org/10.1007/s00013-014-0623-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00013-014-0623-7

Mathematics Subject Classification (2010)

Keywords

Search

Navigation