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


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.


Group ring Unit group Derived length 

Mathematics Subject Classifications (2010)

16S34 16U60 20F16 


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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

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