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

TL;DR

This paper classifies exceptional components in group algebras over abelian number fields, identifying all minimal groups with certain division rings and all finite groups with specific matrix components, impacting unit group constructions.

Contribution

It provides a complete classification of exceptional components in group algebras over abelian number fields, including minimal groups for division rings and all groups with certain matrix components.

Findings

Identified seven minimal groups with division ring components.

Classified all 58 finite groups with faithful exceptional matrix components.

Established infinite classes of division rings for type 1 exceptional components.

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 $FG$.