Abstract :
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 Ω be the Burnside ring functor on G and let ΩF denote the image of the canonical natural transformation Ω → F. Suppose that for each subgroup H subset of(G, every image-torsion element of ΩF(H) is nilpotent. Then the following are equivalent: (1) F is I(G)-hypercomputable ; (2) ΩF is I(G)-hypercomputable ; (3) The restriction homomorphism ResI(G)G(image circle times operator F): image circle times operatorimage F(G) → ∏H set membership, variant I(G) image circle times operatorimage F(H) is injective; (4) the induction homomorphism IndI(G)G (image circle times operator F): coproduct operatorH set membership, variant I(G) image circle times operatorimage F(H) → image circle times operatorimage F(G) is surjective.
