J. Group Theory 13 (2010), 577–588 DOI 10.1515/JGT.2010.008 Journal of Group Theory ( de Gruyter 2010 On the derived length of the unit group of a group algebra Francesco Catino and Ernesto Spinelli (Communicated by F. de Giovanni) Abstract. Let KG be a non-commutative group algebra of a torsion nilpotent group G over a field K of positive characteristic p whose unit group, UðKGÞ, is solvable. In the present note we prove that dlðUðKGÞÞ d dlog 2 ð p þ 1Þe and characterize group algebras for which this lower bound is achieved. 1 Introduction Let KG be the group algebra of a non-abelian group G over a field K of positive char- acteristic p and let UðKGÞ denote its unit group. The investigation of when UðKGÞ is solvable dates back to the beginning of the 1970s with the work by Bateman [3] dealing with finite groups. Necessary and su‰- cient conditions have been recently given by Bovdi [6] in the case in which G contains at least one element of order p (throughout the rest of the paper we refer to these group algebras as modular). By virtue of these results one has that, if p > 3 and G is torsion, then UðKGÞ is solvable if and only if the commutator subgroup G 0 of G is a finite p-group. More conditions are required in the special cases in which p ¼ 2; 3. Though the study of important theoretical properties has been going on for a long time, at the moment very little is known about the derived length of UðKGÞ. Shalev in [14] classified group algebras of finite groups over fields of odd characteristic whose unit group is metabelian, and this classification was completed in even charac- teristic by Kurdics [8] and independently by Coleman and Sandling [7]. Sahai in [13], under the same assumptions as the paper of Shalev, obtained necessary and su‰cient conditions for UðKGÞ to be centrally metabelian. Recently Baginski showed in [1, Theorem 2] that if G is a finite p-group with cyclic commutator subgroup, then dlðUðKGÞÞ ¼ dlog 2 ðjG 0 jþ 1Þe, where the right-hand side of the equality denotes the least integer n with n d log 2 ðjG 0 jþ 1Þ, under the assumption that char K ¼ p d 3. Fi- nally, Balogh and Li [2] computed dlðUðKGÞÞ when G 0 is again a cyclic p-group, for some odd p, but without restrictions on G.