Algebras and Representation Theory

, Volume 17, Issue 5, pp 1597–1601 | Cite as

Group Rings with Solvable Unit Groups of Minimal Derived Length

  • Gregory T. Lee
  • Sudarshan K. Sehgal
  • Ernesto Spinelli
Article
  • 149 Downloads

Abstract

Let F be a field of characteristic p > 2 and G a nonabelian nilpotent group containing elements of order p. Write F G for the group ring. The conditions under which the unit group 𝒰(F G) is solvable are known, but only a few results have been proved concerning its derived length. It has been established that if G is torsion, the minimum derived length is ⌈log2(p + 1)⌉, and this minimum occurs if and only if |G′| = p. In the present note, we show that the same holds if G has elements of infinite order.

Keywords

Group ring Unit group Derived length 

Mathematics Subject Classifications (2010)

16S34 16U60 20F16 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bagiński, C.: A note on the derived length of the unit group of a modular group algebra. Comm. Algebra. 30, 4905–4913 (2002).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balogh, Z., Li, Y.: On the derived length of the group of units of a group algebra. J. Algebra Appl. 6, 991–999 (2007).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bateman, J.M.: On the solvability of unit groups of group algebras. Trans. Amer. Math. Soc. 157, 73–86 (1971).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bovdi, A.: Group algebras with a solvable group of units. Comm. Algebra. 33, 3725–3738 (2005).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Catino, F., Spinelli, E.: On the derived length of the unit group of a group algebra. J. Group Theory. 13, 577–588 (2010).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chandra, H., Sahai, M.: Group algebras with unit groups of derived length three. J. Algebra Appl. 9, 305–314 (2010).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lee, G.T.: Group identities on units and symmetric units of group rings. In: Algebra and Applications, vol. 12. Springer, London (2010).CrossRefGoogle Scholar
  8. 8.
    Passi, I.B.S., Passman, D.S., Sehgal, S.K.: Lie solvable group rings. Canad. J. Math. 25, 748–757 (1973)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Passman, D.S.: Group algebras whose units satisfy a group identity II. Proc. Amer. Math. Soc. 125, 657–662 (1997).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robinson, D.J.S.: A course in the theory of groups, 2nd edn. Springer, New York (1996).CrossRefGoogle Scholar
  11. 11.
    Sehgal, S.K.: Topics in group rings. Dekker, New York (1978).MATHGoogle Scholar
  12. 12.
    Shalev, A.: Meta-abelian unit groups of group algebras are usually abelian. J. Pure Appl. Algebra. 72, 295–302 (1991).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Gregory T. Lee
    • 1
  • Sudarshan K. Sehgal
    • 2
  • Ernesto Spinelli
    • 3
  1. 1.Department of Mathematical SciencesLakehead UniversityThunder BayCanada
  2. 2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  3. 3.Dipartimento di Matematica “G. Castelnuovo”Università degli Studi di Roma “La Sapienza”RomeItaly

Personalised recommendations