Skip to main content
SpringerLink
Log in

An Erdős-Ko-Rado theorem for the group \(\hbox {PSU}(3, q)\)

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper we consider the derangement graph for the group \(\mathop {\text {PSU}}(3,q)\), where q is a prime power. We calculate all eigenvalues for this derangement graph and use the eigenvalues to prove that \(\mathop {\text {PSU}}(3,q)\), under its two-transitive action on a set of size \(q^3+1\), has the Erdős-Ko-Rado property and, provided that \(q\ne 2, 5\), another property that we call the Erdős-Ko-Rado module property.

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. Ahmadi B., Meagher K.: A new proof for the Erdős-Ko-Rado theorem for the alternating group. Discret. Math. 324, 28–40 (2014).

    Article  MATH  Google Scholar 

  2. Ahmadi B., Meagher K.: The Erdős-Ko-Rado property for some \(2\)-transitive groups. Ann. Comb. 19(4), 621–640 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahmadi B., Meagher K.: The Erdős-Ko-Rado property for some permutation groups. Aust. J. Comb. 61, 23–41 (2015).

    MATH  Google Scholar 

  4. Babai L.: Spectra of Cayley graphs. J. Comb. Theory B. 2, 180–189 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  5. Cameron P.J., Ku C.Y.: Intersecting families of permutations. Eur. J. Comb. 24, 881–890 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  6. Diaconis P., Shahshahani M.: Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57(2), 159–179 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  7. Dixon J.D., Mortimer B.: Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York (1996).

    Book  MATH  Google Scholar 

  8. Erdős P., Ko C., Rado R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 12, 313–320 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  9. Godsil C., Meagher K.: A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations. Eur. J. Comb. 30, 404–414 (2009).

    Article  MATH  Google Scholar 

  10. Godsil C., Meagher K.: Erdős-Ko-Rado Theorems: Algebraic Approaches. Cambridge University Press, Cambridge (2015).

    MATH  Google Scholar 

  11. Godsil C., Royle G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001).

    Book  MATH  Google Scholar 

  12. Hoffman AJ.: On eigenvalues and colorings of graphs. In: Graph Theory and Its Applications, (Proc. Adv. Sem., Math. Research Center, Univ. of Wisconsin, Madison, WI, 1969). Academic Press, New York (1970).

  13. Ku C.Y., Wong T.W.H.: Intersecting families in the alternating group and direct product of symmetric groups. Electron. J. Comb. 14, 25 (2007). (electronic).

    MathSciNet  MATH  Google Scholar 

  14. Larose B., Malvenuto C.: Stable sets of maximal size in Kneser-type graphs. Eur. J. Comb. 25, 657–673 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  15. Long L., Plaza R., Sin P.: Qing Xiang characterization of intersecting families of maximum size in \({\mathop {\rm PSL}}(2, q)\). J. Comb. Theory Ser. A 157, 461–499 (2018).

    Article  Google Scholar 

  16. Meagher K., Spiga P.: An Erdős-Ko-Rado theorem for the derangement graph of \(\rm PGL(2, q)\) acting on the projective line. J. Comb. Theory Ser. A 118, 532–544 (2011).

    Article  MATH  Google Scholar 

  17. Meagher K., Spiga P.: An Erdős-Ko-Rado theorem for the derangement graph of \(\rm PGL_3(q)\) acting on the projective plane. SIAM J. Discret. Math. 28, 918–941 (2011).

    Article  MATH  Google Scholar 

  18. Meagher K., Spiga P.: Pham Huu Tiep, An Erdős-Ko-Rado theorem for finite 2-transitive groups. Eur. J. Comb. 55, 100–118 (2016).

    Article  MATH  Google Scholar 

  19. Simpson W.A., Frame S.J.: The character tables for \({\rm SL}(3,\, q)\), \({\rm SU}(3,\, q^{2})\), \({\rm PSL}(3,\, q)\), \({\rm PSU}(3,\, q^{2})\). Can. J. Math. 25, 486–494 (1973).

    Article  MATH  Google Scholar 

  20. The GAP Group, GAP—Groups.: Algorithms, and Programming, Version 4.4.12; (http://www.gap-system.org) (2008).

Download references

Acknowledgements

The author would like to thank the anonymous referees who checked this paper in detail and offered excellent suggests to improve it.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Karen Meagher

Authors
  1. Karen Meagher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karen Meagher.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Finite Geometries”.

Research supported in part by an NSERC Discovery Research Grant, Application No.: RGPIN-341214-2013.

Appendix

Appendix

See Tables 10 and 11.

Table 10 Partial character table for \(\mathop {\text {PSU}}(3,q)\) where \(3 \!\! \not | \,q+1\)
Full size table
Table 11 Partial character table for \(\mathop {\text {PSU}}(3,q)\) where \(3 | q+1\)
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meagher, K. An Erdős-Ko-Rado theorem for the group \(\hbox {PSU}(3, q)\). Des. Codes Cryptogr. 87, 717–744 (2019). https://doi.org/10.1007/s10623-018-0537-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-018-0537-7

Keywords

Mathematics Subject Classification

Search

Navigation