The Relative Burnside Kernel - The Elementary Abelian Case

TL;DR

This paper investigates the kernel of a map from finite G×Z_p-sets to rational permutation modules, providing a conjectural description and proving it for elementary abelian and cyclic p-groups.

Contribution

It offers a conjectural framework for the kernel in the elementary abelian case and proves it for specific p-groups, advancing understanding of permutation modules.

Findings

Conjectural description of the kernel for finite G×Z_p-sets.

Proof of the conjecture for elementary abelian p-groups.

Proof of the conjecture for cyclic p-groups.

Abstract

We give a conjectural description for the kernel of the map assigning to each finite $\mathbb Z_p$-free $G\times\mathbb Z_p$-set its rational permutation module where G is a finite p-group. We prove that this conjecture is true when G is an elementary abelian p-group or a cyclic p-group.