Induction for Finite-Groups Revisited

Bak A. Induction for Finite-Groups Revisited. Journal of Pure and Applied Algebra. 1995;104(3):235-241.Let G be a finite group and I(G) a nonempty family of subgroups of G, which is closed under conjugation and taking subgroups of its members. Let F be a Green ring functor on G. Let a be the Burnside ring functor on G and let Q(F), denote the image of the canonical natural transformation Omega --> F. Suppose that for each subgroup H subset of G, every Z-torsion element of Omega(F)(H) is nilpotent. Then the following are equivalent: (1) F is I(G)-hypercomputable; (2) Omega(F) is I(G)-hypercomputable; (3) The restriction homomorphism Res(I(G))(G) (Q x F): Q x(z) F(G) --> Pi(H is an element of I(G)) Q x(z) F(H) is injective; (4) the induction homomorphism Ind(I(G))(G) (Q x F):coproduct (H is an element of I(G)) Q x(z) F(H) --> Q x(z) F(G) is surjective