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.
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References
Bagiński, C.: A note on the derived length of the unit group of a modular group algebra. Comm. Algebra. 30, 4905–4913 (2002).
Balogh, Z., Li, Y.: On the derived length of the group of units of a group algebra. J. Algebra Appl. 6, 991–999 (2007).
Bateman, J.M.: On the solvability of unit groups of group algebras. Trans. Amer. Math. Soc. 157, 73–86 (1971).
Bovdi, A.: Group algebras with a solvable group of units. Comm. Algebra. 33, 3725–3738 (2005).
Catino, F., Spinelli, E.: On the derived length of the unit group of a group algebra. J. Group Theory. 13, 577–588 (2010).
Chandra, H., Sahai, M.: Group algebras with unit groups of derived length three. J. Algebra Appl. 9, 305–314 (2010).
Lee, G.T.: Group identities on units and symmetric units of group rings. In: Algebra and Applications, vol. 12. Springer, London (2010).
Passi, I.B.S., Passman, D.S., Sehgal, S.K.: Lie solvable group rings. Canad. J. Math. 25, 748–757 (1973)
Passman, D.S.: Group algebras whose units satisfy a group identity II. Proc. Amer. Math. Soc. 125, 657–662 (1997).
Robinson, D.J.S.: A course in the theory of groups, 2nd edn. Springer, New York (1996).
Sehgal, S.K.: Topics in group rings. Dekker, New York (1978).
Shalev, A.: Meta-abelian unit groups of group algebras are usually abelian. J. Pure Appl. Algebra. 72, 295–302 (1991).
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The first named author was supported by “Programma Professori Visitatori 2013” of GNSAGA. This research was also supported by the Università degli Studi di Roma “La Sapienza” and NSERC of Canada. The article was written while the first two named authors were visiting the Università degli Studi di Roma “La Sapienza”.
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Lee, G.T., Sehgal, S.K. & Spinelli, E. Group Rings with Solvable Unit Groups of Minimal Derived Length. Algebr Represent Theor 17, 1597–1601 (2014). https://doi.org/10.1007/s10468-013-9461-8
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DOI: https://doi.org/10.1007/s10468-013-9461-8