2014/12/31 by Andreas Bächle, Mauricio Caicedo, Inneke Van Gelder
Mathematics · #Abelian group #Algebraic Geometry and Number Theory #Algebraic number field #Division (mathematics) #Division ring #Field (mathematics) #Finite Group Theory Research #Finite group #Group (periodic table) #Rings, Modules, and Algebras #Type (biology) #Unit (ring theory) #math.GR #math.RA #math.RT #msc:16G30 #msc:16K20 #msc:16S34 #msc:19B37 #msc:20C05
paper · pdf · doi:10.1142/s0219498816500924
published as Journal of Algebra and Its Applications 15 (2016), no. 5, 1650092, 32 pages · 23 pages, [v4]: introduction and motivation has been changed, typos corrected
arxiv created 2015/03/09 · openalex publication_date 2015/06/08 · arxiv updated 2016/06/07 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
When considering the unit group of [Formula: see text] ([Formula: see text] the ring of integers of an abelian number field [Formula: see text] and a finite group [Formula: see text]) certain components in the Wedderburn decomposition of [Formula: see text] cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 is division rings, type 2 is [Formula: see text]-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (with respect to quotients) having those division rings in their Wedderburn decomposition over [Formula: see text]. We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields [Formula: see text] by describing all 58 finite groups [Formula: see text] having a faithful exceptional Wedderburn component of this type in [Formula: see text].