J. Group Theory 13 (2010), 221–231 DOI 10.1515/JGT.2009.048 Journal of Group Theory ( de Gruyter 2010 Group algebras of torsion groups and Lie nilpotence A. Giambruno, C. Polcino Milies and Sudarshan K. Sehgal (Communicated by S. Sidki) Abstract. Let  be an involution of a group algebra FG induced by an involution of the group G. For char F 0 2, we classify the torsion groups G with no elements of order 2 whose Lie al- gebra of -skew elements is nilpotent. 1 Introduction Let F be a field and FG the group algebra of a group G over F . If  is an involution of FG then the set of skew elements FG  ¼fx A FG j x  ¼xg is a Lie algebra. Here we are interested in classifying the groups G for which such an algebra is nilpotent. We shall assume throughout that char F 0 2. A natural involution of FG to consider is the so-called classical involution, obtained by linearly extending the group involu- tion g ! g 1 to FG. For this involution the problem has been completed settled in [4] for groups with no elements of order 2 and in [5] for arbitrary groups. We should mention that if we regard FG as a Lie algebra under the usual Lie bracket, then from results in [9] the algebra FG is nilpotent if and only if either char F ¼ 0 and G is abelian or char F ¼ p > 0 and G is a nilpotent group whose de- rived group is a finite p-group. The same classification holds if G has no elements of order 2 and we only impose that FG  is Lie nilpotent under the classical involution; see [4]. Here we try to extend this result to an involution of FG obtained as a linear exten- sion of a group involution of G. We shall classify the groups G for which FG  is Lie nilpotent when G is a torsion group and has no elements of order 2. It turns out that the conclusion is much more involved than for the classical involution. Our main re- sult is the following. The first author was partially supported by MIUR of Italy. The second author was partially supported by FAPESP, Proc. 2005/60411-8 and CNPq., Proc. 300243/79-0(RN) of Brazil. The third author was supported by NSERC of Canada.