Brauer relations in finite groups

TL;DR

This paper classifies Brauer relations, which are elements in the kernel of the map from the Burnside ring to the representation ring, for all finite groups, extending previous classifications for p-groups.

Contribution

It extends the classification of Brauer relations from p-groups to all finite groups, providing a comprehensive understanding of these relations.

Findings

Classification of Brauer relations for all finite groups

Extension of previous p-group results

Identification of kernels in the Burnside to representation ring map

Abstract

If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the representation ring of G has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave-Bouc classification in the case of p-groups.