Locally solvable unit group of crossed products

摘要

Let F be a field of characteristic p >= 0, with vertical bar F vertical bar > 3, and G be a torsion group. We provide some necessary conditions for the unit group of a crossed product F * G to be locally solvable or locally nilpotent. As some special cases of crossed products, let F(t)G and F(sigma)G denote a twisted group algebra and skew group algebra, respectively. In this paper, among other results, we show that U(F(t)G) is a locally solvable group if and only if G is locally solvable, G' is a p-group and F(t)G is stably untwisted. Also, U(F(sigma)G) is a locally nilpotent (solvable, if p not equal 2, 3) if and only if G is locally nilpotent (solvable), G' is a p-group and r is trivial. As a special result, for finite group G, it is shown that U (F(t)G) is a nilpotent group if and only if F(t)G is a Lie nilpotent ring.