Locally solvable unit group of crossed products

Abstract

Let F be a field of characteristic with and G be a torsion group. We provide some necessary conditions for the unit group of a crossed product to be locally solvable or locally nilpotent. As some special cases of crossed products, let and denote a twisted group algebra and skew group algebra, respectively. In this paper, among other results, we show that is a locally solvable group if and only if G is locally solvable, is a p-group and is stably untwisted. Also, is a locally nilpotent (solvable, if ) if and only if G is locally nilpotent (solvable), is a p-group and σ is trivial. As a special result, for finite group G, it is shown that is a nilpotent group if and only if is a Lie nilpotent ring.