Skip to main content

\(\varvec{z}\)-Classes in finite groups of conjugate type \(\varvec{(n,1)}\)

Abstract

Two elements in a group G are said to be z-equivalent or to be in the same z-class if their centralizers are conjugate in G. In a recent work, Kulkarni et al. (J. Algebra Appl., 15 (2016) 1650131) proved that a non-abelian p-group G can have at most \(\frac{p^k-1}{p-1} +1\) number of z-classes, where \(|G/Z(G)|=p^k\). Here, we characterize the p-groups of conjugate type (n, 1) attaining this maximal number. As a corollary, we characterize p-groups having prime order commutator subgroup and maximal number of z-classes.

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

References

  1. Bose A, On the genus number of algebraic groups, J. Ramanujan Math. Soc., 28(4) (2013) 443–482

    MathSciNet  Google Scholar 

  2. Carter R W, Centralizers of semisimple elements in the finite classical groups, Proc. London Math. Soc. (3), 42(1) (1981) 1–41

    MathSciNet  Article  MATH  Google Scholar 

  3. Craven D A, The theory of \(p\)-groups, http://web.mat.bham.ac.uk/D.A.Craven/pgroups.html (2008)

  4. Gongopadhyay K and Kulkarni R S, \(z\)-Classes of isometries of the hyperbolic space, Conform. Geom. Dyn., 13 (2009) 91–109

    MathSciNet  Article  MATH  Google Scholar 

  5. Gongopadhyay K and Kulkarni R S, The \(Z\)-classes of isometries, J. Indian Math. Soc. (N.S.), 81(3–4) (2014) 245–258

    MathSciNet  MATH  Google Scholar 

  6. Gongopadhyay K, The \(z\)-classes of quaternionic hyperbolic isometries, J. Group Theory, 16(6) (2013) 941–964

    MathSciNet  Article  MATH  Google Scholar 

  7. Gouraige R, \(Z\)-classes in central simple algebras, ProQuest LLC (2006) (MI: Ann Arbor) Ph.D. thesis, City University of New York

  8. Hall P, The classification of prime-power groups, J. Reine Angew. Math., 182 (1940) 130–141

    MathSciNet  MATH  Google Scholar 

  9. Itô N, On finite groups with given conjugate types I, Nagoya Math. J., 6 (1953) 17–28

    MathSciNet  Article  MATH  Google Scholar 

  10. Kitture R D, \(z\)-Classes in finite \(p\)-groups Ph.D. thesis (2014) (University of Pune)

  11. Kulkarni R, Kitture R D and Jadhav V S, \(z\)-Classes in groups, J. Algebra Appl., 15 (2016) 1650131, 16d pp., https://doi.org/10.1142/S0219498816501310.

  12. Kulkarni R S, Dynamical types and conjugacy classes of centralizers in groups, J. Ramanujan Math. Soc., 22(1) (2007) 35–56

    MathSciNet  MATH  Google Scholar 

  13. Kulkarni R S, Dynamics of linear and affine maps, Asian J. Math., 12(3) (2008) 321–343

    MathSciNet  Article  MATH  Google Scholar 

  14. Singh A, Conjugacy classes of centralizers in \(G_2\), J. Ramanujan Math. Soc., 23(4) (2008) 327–336

    MathSciNet  MATH  Google Scholar 

  15. Steinberg R, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, vol. 366 (1974) (Berlin-New York: Springer-Verlag), notes by Vinay V Deodhar

  16. Wilson R A, The finite simple groups, vol. 251 of Graduate Texts in Mathematics (2009) (London: Springer-Verlag London Ltd)

Download references

Acknowledgements

The authors are thankful to Rahul Kitture for letting them know about his work and for many comments on this work. The authors are grateful to Silvio Dolfi for useful comments and suggestions. This work was part of the MS thesis of Shivam Arora at IISER, Mohali. He gratefully acknowledges the support of IISER Mohali during the course of this work. The second author, Gongopadhyay acknowledges partial support from SERB-DST Grant SR/FTP/MS-004/2010.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Krishnendu Gongopadhyay.

Additional information

Communicating Editor: B Sury

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arora, S., Gongopadhyay, K. \(\varvec{z}\)-Classes in finite groups of conjugate type \(\varvec{(n,1)}\). Proc Math Sci 128, 31 (2018). https://doi.org/10.1007/s12044-018-0412-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12044-018-0412-5

Keywords

  • Conjugacy classes of centralizers
  • z-classes
  • p-groups
  • extraspecial groups

2010 Mathematics Subject Classification

  • Primary: 20D15
  • Secondary: 20E45