Relative Brauer relations of abelian p-groups

TL;DR

This paper extends the classification of relative Brauer relations from elementary abelian p-groups to all finite abelian p-groups, connecting group actions, bisets, and stable homotopy theory.

Contribution

It generalizes Kahn's classification of relative Brauer relations for elementary abelian p-groups to all finite abelian p-groups.

Findings

Extended classification of relative Brauer relations to all finite abelian p-groups.

Connected Brauer relations with stable homotopy theory.

Provided a comprehensive framework for $(G,C_p)$-bisets in the abelian p-group context.

Abstract

The Brauer relations of a finite group $G$ are virtual differences of non-isomorphic $G$-sets $X-Y$ which induce isomorphic permutation $G$-representations $\mathbb Q[X]\simeq\mathbb Q[Y]$ over the rationals. These relations have been classified by Tornehave-Bouc and Bartel-Dokchitser. Motivated by stable homotopy theory, a relative version of Brauer relations for $(G,C_p)$-bisets which are $C_p$-free have been classified by Kahn in case $G$ is an elementary Abelian $p$-group. In this paper we extend Kahn's classification to the case when $G$ is a finite Abelian $p$-group.