Locally solvable unit group of crossed products. Issue 12 (1st December 2020)

• Record Type: Journal Article
• Authors: Ramezan-Nassab, M.; Bien, M. H.
• Is Part Of: Communications in algebra. Volume 48:Issue 12(2020)
• Page Start: 5247
• Page End: 5253
• Publication Date: 2020-12-01
• Subjects: Division ring -- Lie nilpotent -- skew group ring -- twisted group ring; 16K40 -- 16R40 -- 16S34 -- 16S35; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2020.1784911
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: Let F be a field of characteristic p ≥ 0, with | F | > 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 τ G and F σ G denote a twisted group algebra and skew group algebra, respectively. In this paper, among other results, we show that U ( F τ G ) is a locally solvable group if and only if G is locally solvable, G ′ is a p -group and F τ G is stably untwisted. Also, U ( F σ G ) is a locally nilpotent (solvable, if p ≠ 2, 3 ) if and only if G is locally nilpotent (solvable), G ′ is a p -group and σ is trivial. As a special result, for finite group G, it is shown that U ( F τ G ) is a nilpotent group if and only if F τ G is a Lie nilpotent ring.