Lie properties of symmetric elements in group rings II

Abstract

Let F be a field of characteristic different from 2, and G a group with involution ∗ . Write ( F G ) + for the set of elements in the group ring F G that are symmetric with respect to the induced involution. Recently, Giambruno, Polcino Milies and Sehgal showed that if G has no 2-elements, and ( F G ) + is Lie nilpotent (resp. Lie n -Engel), then F G is Lie nilpotent (resp. Lie m -Engel, for some m ). Here, we classify the groups containing 2-elements such that ( F G ) + is Lie nilpotent or Lie n -Engel.