which acts on the first factor: for all Properties The results introduced in this section will be presented without proof. These may be found in [1] and [2]. Uniqueness and existence of the induced representation. Let be a linear representation of a subgroup of Then there exists a linear representation of which is induced by Note that this representation is unique up to isomorphism. Transitivity of induction. Let be a representation of and let be an ascending series of groups. Then we have Lemma. Let be induced by and let be a linear representation of Now let be a linear map satisfying the property that for all Then there exists a uniquely determined linear map which extends and for which is valid for all This means that if we interpret as a –module, we have where is the vector space of all –homomorphisms of to The same is valid for Induction on class functions. In the same way as it was done with representations, we can - by induction - obtain a class function on the group from a class function on a subgroup. Let be a class function on We define a function on by We say is induced by and write or Proposition. The function is a class function on If is the character of a representation of then is the character of the induced representation of Lemma. If is a class function on and is a class function on then we have: Theorem. Let be the representation of induced by the representation of the subgroup Let and be the corresponding characters. Let be a representative system of The induced character is given by Frobenius reciprocity As a preemptive summary, the lesson to take from Frobenius reciprocity is that the maps and are adjoint to each other. Let be an irreducible representation of and let be an irreducible representation of then the Frobenius reciprocity tells us that is contained in as often as is contained in Frobenius reciprocity. If and we have This statement is also valid for the inner product. Mackey's irreducibility criterion George Mackey established a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation. Two representations and of a group are called disjoint, if they have no irreducible component in common, i.e. if Let be a group and let be