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

Abstract

When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ 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 are division rings, type 2 are $2 \times 2$-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. 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 $FG$.