If G is a finite group and k a commutative ring, J. A. Green defines a G- functor A over k to be a family of k-algebras \(A(H)\), with H running over the set of subgroups of G, together with restriction maps \(r^ K_ H: A(K)\to A(H)\), transfer maps \(t^ K_ H: A(H)\to A(K)\) (with \(H\leq K)\) and conjugation maps \(A(H)\to A(gHg^{-1})\), such that certain axioms are satisfied (included Mackey’s formula and Frobenius reciprocity). The Brauer morphism \(br^ G_ H: A(G)\to \bar A(H)=A(H)/\sum_{S<H}t^ H_ S(A(S))\) is by definition the composition of the restriction \(r^ G_ H\) and the canonical map \(br^ H_ H: A(H)\to \bar A(H)\). The set \({\mathcal P}(A)\) of subgroups H of G such that \(\bar A(H)\neq 0\) is shown to be a refinement of the concept of defect base \({\mathcal D}(A)\) introduced by Green: \({\mathcal D}(A)\) is the subgroup closure of \({\mathcal P}(A).\)
The product of all Brauer morphisms \(\beta_ G: A(G)\to (\prod_{H}\bar A(H))^ G\) terminates in the G-fixed points of \(\Pi_ H\bar A(H)\). The main result of the paper says on one hand that both Ker\(\beta_ G\) and Coker \(\beta_ G\) are annihilated by a power of \(| G|\) and on the other hand that Ker\(\beta_ G\) is a nilpotent ideal of \(A(G)\). In particular \(\beta_ G\) is an isomorphism if \(| G|\) is invertible in k and this fact is exploited in various ways. In particular the universal role of the Burnside ring G-functor is used to derive an idempotent formula and an induction formula generalizing known results. Also the notion of symmetric G-functor is defined and analyzed. The character ring and the Green ring are two important examples which are discussed in the light of the results of the paper.