possible to write any character as a sum of irreducible characters. The inner product defined above can be extended on the set of all -valued functions on a finite group: A symmetric bilinear form can also be defined on These two forms match on the set of characters. If there is no danger of confusion the index of both forms and will be omitted. Let be two –modules. Note that –modules are simply representations of . Since the orthonormal property yields the number of irreducible representations of is exactly the number of its conjugacy classes, then there are exactly as many simple –modules (up to isomorphism) as there are conjugacy classes of We define in which is the vector space of all –linear maps. This form is bilinear with respect to the direct sum. In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations. For instance, let and be the characters of and respectively. Then It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations. Theorem. Let be a linear representation of with character Let where are irreducible. Let be an irreducible representation of with character Then the number of subrepresentations which are isomorphic to is independent of the given decomposition and is equal to the inner product i.e. the –isotype of is independent of the choice of decomposition. We also get: and thus Corollary. Two representations with the same character are isomorphic. This means that every representation is determined by its character. With this we obtain a very useful result to analyse representations: Irreducibility criterion. Let be the character of the representation then we have The case holds if and only if is irreducible. Therefore, using the first theorem, the characters of irreducible representations of form an orthonormal set on with respect to this inner product. Corollary. Let be a vector space with A given irreducible representation of is contained –times in the regular representation. In other words, if denotes the regular representation of then we have: in which is the set of all irreducible representations of that are pairwise not isomorphic to each other. In terms of the group algebra, this means that as algebras. As a numerical result we get: in which is the regular representation and and are corresponding characters to