A classification of exceptional components in group algebras over abelian number fields

When considering the unit group of OFG (OF the ring of integers of an abelian number field F and a finite group G) certain components in the Wedderburn decomposition of FG 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 2 × 2-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 F. We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields F by describing all 58 finite groups G having a faithful exceptional Wedderburn component of this type in FG.

Publication year:2015

Keywords:group rings, Wedderburn decomposition, exceptional components, finite groups, abelian number fields

CSS-citation score:1