The Relative Burnside Kernel - The Elementary Abelian Case

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.