LIE METABELIAN SKEW ELEMENTS IN GROUP RINGS

Let F be a field of characteristic p ≠ 2 and G a group without 2-elements having an involution ∗. Extend the involution linearly to the group ring FG, and let (FG)− denote the set of skew elements with respect to ∗. In this paper, we show that if G is finite and (FG)− is Lie metabelian, then G is nilpotent. Based on this result, we deduce that if G is torsion, p > 7 and (FG)− is Lie metabelian, then G must be abelian. Exceptions are constructed for smaller values of p.