class functions defines a homomorphism of abelian groups which will be written as or in short According to the Frobenius reciprocity, these two homomorphisms are adjoint with respect to the bilinear forms and Furthermore, the formula shows that the image of is an ideal of the ring By the restriction of representations, the map can be defined analogously for and by the induction we obtain the map for Due to the Frobenius reciprocity, we get the result that these maps are adjoint to each other and that the image is an ideal of the ring If is a commutative ring, the homomorphisms and may be extended to –linear maps: in which are all the irreducible representations of up to isomorphism. With we obtain in particular that and supply homomorphisms between and Let and be two groups with respective representations and Then, is the representation of the direct product as was shown in a previous section. Another result of that section was that all irreducible representations of are exactly the representations where and are irreducible representations of and respectively. This passes over to the representation ring as the identity in which is the tensor product of the representation rings as –modules. Induction theorems Induction theorems relate the representation ring of a given finite group G to representation rings of a family X consisting of some subsets H of G. More precisely, for such a collection of subgroups, the induction functor yields a map ; induction theorems give criteria for the surjectivity of this map or closely related ones. Artin's induction theorem is the most elementary theorem in this group of results. It asserts that the following are equivalent: The cokernel of is finite. is the union of the conjugates of the subgroups belonging to i.e. Since is finitely generated as a group, the first point can be rephrased as follows: For each character of there exist virtual characters and an integer such that gives two proofs of this theorem. For example, since G is the union of its cyclic subgroups, every character of is a linear combination with rational coefficients of characters induced by characters of cyclic subgroups of Since the representations of cyclic groups are well-understood, in particular the irreducible representations are one-dimensional, this gives a certain control over representations of G. Under the above circumstances, it is not in general true that is surjective. Brauer's induction theorem asserts that