The relative Burnside kernel: the elementary abelian case (Q350780)

Let \(p\) be a prime, and let \(G,H\) be finite \(p\)-groups. The paper under review is concerned with the \(H\)-free double Burnside group \(A(G,H)\) and the \(H\)-free double representation group \(R(G,H)\). Recall that \(A(G,H)\) is generated by the isomorphism classes of finite \(G\)-\(H\)-bisets on which \(H\) acts freely, and \(R(G,H)\) is generated by the isomorphism classes of finitely generated \(\mathbb{Q}G\)-\(\mathbb{Q}H\)-bimodules on which \(\mathbb{Q}H\) acts freely. The author denotes the kernel of the linearization map \(f:A(G,H) \to R(G,H)\) by \(N(G,H)\). If \(|H|=p\) then \(f\) is known to be surjective. The author states a conjecture on generators of \(N(G,H)\) and proves this conjecture in the case where \(G\) is elementary abelian or cyclic. NEWLINENEWLINENEWLINEReviewer's remark: In this context, Chapter 11 of \textit{S. Bouc}'s book [Biset functors for finite groups. Berlin: Springer (2010; Zbl 1205.19002)] appears to be relevant.