a subgroup. We define for Let be a representation of the subgroup This defines by restriction a representation of We write for We also define another representation of by These two representations are not to be confused. Mackey's irreducibility criterion. The induced representation is irreducible if and only if the following conditions are satisfied: is irreducible For each the two representations and of are disjoint. For the case of normal, we have and . Thus we obtain the following: Corollary. Let be a normal subgroup of Then is irreducible if and only if is irreducible and not isomorphic to the conjugates for Applications to special groups In this section we present some applications of the so far presented theory to normal subgroups and to a special group, the semidirect product of a subgroup with an abelian normal subgroup. Proposition. Let be a normal subgroup of the group and let be an irreducible representation of Then one of the following statements has to be valid: either there exists a proper subgroup of containing , and an irreducible representation of which induces , or is an isotypic -module. Proof. Consider as an -module, and decompose it into isotypes as . If this decomposition is trivial, we are in the second case. Otherwise, the larger -action permutes these isotypic modules; because is irreducible as a -module, the permutation action is transitive (in fact primitive). Fix any ; the stabilizer in of is elementarily seen to exhibit the claimed properties. Note that if is abelian, then the isotypic modules of are irreducible, of degree one, and all homotheties. We obtain also the following Corollary. Let be an abelian normal subgroup of and let be any irreducible representation of We denote with the index of in Then [1] If is an abelian subgroup of (not necessarily normal), generally is not satisfied, but nevertheless is still valid. Classification of representations of a semidirect product In the following, let be a semidirect product such that the normal semidirect factor, , is abelian. The irreducible representations of such a group can be classified by showing that all irreducible representations of can be constructed from certain subgroups of . This is the so-called method of “little groups” of Wigner and Mackey. Since is abelian, the irreducible characters of have degree one and form the group The group acts on by for Let be a representative system of the orbit