Skip to main content

Constructive Representation Theoretic Methods and Non-Abelian Difference Sets

  • Chapter
Difference Sets, Sequences and their Correlation Properties

Part of the book series: NATO Science Series ((ASIC,volume 542))

Abstract

The historical development of a mathematical subject has often begun with the commutative case and then built on this case to unravel the non-commutative. Although the role of non-abelian groups in algebraic combinatorics and finite geometry goes back at least to Dickson (1901), genuinely non-abelian difference sets have only appeared in the last few years, see Liebler and Smith (1993), Smith (1995).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Assmus, E.F.Jr. and Key, J.D. (1992) Designs and Their Codes, Cambridge University Press.

    Google Scholar 

  • Brauer, R. (1945) On the representation of a group of order g in the field of th g-th roots of unity, Amer. J. Math. 67, 243–250.

    Article  MathSciNet  Google Scholar 

  • Bruck, R. C. (1955) Difference sets in a finite group, Trans. Amer. Math. Soc. 78, 464–481.

    Article  MathSciNet  MATH  Google Scholar 

  • Curtis, C.W. and Reiner, I. (1966) Representation Theory of Finite Groups and Associative Algebras, J. Wiley, New York.

    Google Scholar 

  • Davis, J.A. and Iiams, J.E. (1998) Hadamard difference sets in non-abelian 2-groups with high exponent, J. Algebra 199, 62–87.

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, J.A. and Smith, K.W. (1994) A construction of difference sets in high exponent 2-groups using representation theory, J. Algebraic Combin. 3, 137–151.

    Article  MathSciNet  MATH  Google Scholar 

  • Dickson, L.E. (1901) Linear Groups with an Exposition of the Galois Field Theory, Teubner, Leipzig (reprint Dover, New York 1958).

    Google Scholar 

  • Frobenius, F.G. (1968) Gesammelte Abhandlungen 3, Springer, Berlin.

    Google Scholar 

  • Hawkins, T. (1974) New light on Frobenius’ creation of the theory of group characters, Arch. History Exact Sci. 12, 217–243.

    Article  MathSciNet  MATH  Google Scholar 

  • Hughes, D.R. (1957) Generalized incidence matrices over group algebras, Ill. J. Math. 1, 545–551.

    MATH  Google Scholar 

  • Isaacs, I.M. (1976) Character Theory of Finite Groups, Academic Press, New York.

    MATH  Google Scholar 

  • James, G. and Kerber, A. (1981) The Representation Theory of the Symmetric Group, Addison Wesley, London.

    MATH  Google Scholar 

  • Jungnickel, D. and Pott, A. (1999) Difference sets: an introduction, the volume.

    Google Scholar 

  • Liebler, R.A. (1981) Relations among the projective geometry codes, in Finite Geometries and Designs, London Math. Soc. Lecture Notes Series 49, pp.221–225.

    MathSciNet  Google Scholar 

  • Liebler, R.A. (1993) The inversion formula, J. Combin. Math. and Combin. Computing 13, 143–160.

    MathSciNet  MATH  Google Scholar 

  • Liebler, R.A. and Mena, R.A. (1988) Certain distance-regular digraphs and related rings of characteristic 4, Journal of Comb. Theory (A) 47, 111– 123.

    Article  MathSciNet  Google Scholar 

  • Liebler, R.A. and Smith, K.W. (1993) On difference sets in certain 2-groups, in D. Jungnickel (ed.), Coding Theory, Design Theory,Group Theory: Proceedings of the Marshall Hall Conference, John Wiley & Sons, New York.

    Google Scholar 

  • Maranda, J.M. (1953) On p-adic integral representations of finite groups, Canad. J. Math. 5,344–355.

    Article  MathSciNet  MATH  Google Scholar 

  • McFarland, R.L. (1989) Difference sets in abelian groups of order 4p2, Mitt. Math. Sem. Giessen 192, 1–70.

    MathSciNet  Google Scholar 

  • Smith, K.W. (1995) Non abelian Hadamard difference sets, Journal of Comb. Theory (A) 70, 144–156.

    Article  MATH  Google Scholar 

  • Turyn, R. (1965) Character Sums and Difference Sets, Pac. J. Math. 15, 319–346.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Liebler, R.A. (1999). Constructive Representation Theoretic Methods and Non-Abelian Difference Sets. In: Pott, A., Kumar, P.V., Helleseth, T., Jungnickel, D. (eds) Difference Sets, Sequences and their Correlation Properties. NATO Science Series, vol 542. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4459-9_13

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4459-9_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-5959-3

  • Online ISBN: 978-94-011-4459-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics