Group Algebras with Locally Nilpotent Unit Groups

Abstract

Let F be a field of characteristic p ≥ 0 and G any group. The local nilpotency of the group of units of the group algebra FG is investigated. We show that if 𝒰(FG) is locally nilpotent, then the set of p-elements of G form a subgroup P and the torsion elements of G/P form an abelian group. If, in addition, the set of nilpotent elements of FG is finite, every idempotent in F(G/P) is central; a converse version is also indicated. As a result, we show that, if G is torsion, then 𝒰(FG) is locally nilpotent if and only if G is locally nilpotent and G′ is a p-group, if and only if FG is Lie Engel and G is locally finite.