Groups with a nontrivial nonideal kernel

TL;DR

This paper classifies finite groups based on the structure of their rational group algebra, specifically those that decompose into a product involving a proper factor group and division rings.

Contribution

It provides a classification of finite groups whose rational group algebra decomposes into a product of a smaller group's algebra and division rings, revealing new structural insights.

Findings

Characterization of groups with such algebra decompositions

Identification of conditions for the algebra to split as described

Extension of known classifications in group algebra theory

Abstract

We classify finite groups $G$, such that the group algebra, $\mathbb{Q}G$ (over the field of rational numbers $\mathbb{Q}$), is the direct product of the group algebra $\mathbb{Q}[G/N]$ of a proper factor group $G/N$, and some division rings.