Skip to main content
SpringerLink
Log in

Unitary units in group algebras

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

LetK[G] denote the group algebra of the finite groupG over the non-absolute fieldK of characteristic ≠ 2, and let *:K[G] →K[G] be theK-involution determined byg*=g −1 for allg ∈G. In this paper, we study the group A = A(K[G]) of unitary units ofK[G] and we classify those groupsG for which A contains no nonabelian free group. IfK is algebraically closed, then this problem can be effectively studied via the representation theory ofK[G]. However, for general fields, it is preferable to take an approach which avoids having to consider the division rings involved. Thus, we use a result of Tits to construct fairly concrete free generators in numerous crucial special cases.

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. B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

    MATH  Google Scholar 

  2. I. M. Isaacs,Character Theory of Finite Groups, Academic Press, New York, 1976.

    MATH  Google Scholar 

  3. I. M. Isaacs,Algebra: A Graduate Course, Brooks/Cole, Pacific Grove, 1994.

    Google Scholar 

  4. D. S. Passman,The Algebraic Structure of Group Rings, Wiley-Interscience, New York, 1977.

    MATH  Google Scholar 

  5. I. N. Sanov,A property of a representation of a free group, Doklady Akademii Nauk SSSR57 (1947), 657–659 (Russian).

    MATH  MathSciNet  Google Scholar 

  6. J. Tits,Free subgroups in linear groups, Journal of Algebra20 (1972), 250–270.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of São Paulo, 05389-970, São Paulo, Brazil

    J. Z. Gonçalves

  2. Department of Mathematics, University of Wisconsin, 53706, Madison, WI, USA

    D. S. Passman

Authors
  1. J. Z. Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. S. Passman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Z. Gonçalves.

Additional information

The first author’s research was supported in part by Capes and Fapesp - Brazil.

The second author’s research was supported in part by NSF Grant DMS-9224662.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gonçalves, J.Z., Passman, D.S. Unitary units in group algebras. Isr. J. Math. 125, 131–155 (2001). https://doi.org/10.1007/BF02773378

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02773378

Keywords

Search

Navigation