of in For every let This is a subgroup of Let be the corresponding subgroup of We now extend the function onto by for Thus, is a class function on Moreover, since for all it can be shown that is a group homomorphism from to Therefore, we have a representation of of degree one which is equal to its own character. Let now be an irreducible representation of Then we obtain an irreducible representation of by combining with the canonical projection Finally, we construct the tensor product of and Thus, we obtain an irreducible representation of To finally obtain the classification of the irreducible representations of we use the representation of which is induced by the tensor product Thus, we achieve the following result: Proposition. is irreducible. If and are isomorphic, then and additionally is isomorphic to Every irreducible representation of is isomorphic to one of the Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in [1]. In other words, we classified all irreducible representations of Representation ring The representation ring of is defined as the abelian group With the multiplication provided by the tensor product, becomes a ring. The elements of are called virtual representations. The character defines a ring homomorphism in the set of all class functions on with complex values in which the are the irreducible characters corresponding to the Because a representation is determined by its character, is injective. The images of are called virtual characters. As the irreducible characters form an orthonormal basis of induces an isomorphism This isomorphism is defined on a basis out of elementary tensors by respectively and extended bilinearly. We write for the set of all characters of and to denote the group generated by i.e. the set of all differences of two characters. It then holds that and Thus, we have and the virtual characters correspond to the virtual representations in an optimal manner. Since holds, is the set of all virtual characters. As the product of two characters provides another character, is a subring of the ring of all class functions on Because the form a basis of we obtain, just as in the case of an isomorphism Let be a subgroup of The restriction thus defines a ring homomorphism which will be denoted by or Likewise, the induction on