and respectively. Recall that denotes the neutral element of the group. This formula is a "necessary and sufficient" condition for the problem of classifying the irreducible representations of a group up to isomorphism. It provides us with the means to check whether we found all the isomorphism classes of irreducible representations of a group. Similarly, by using the character of the regular representation evaluated at we get the equation: Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations: The Fourier inversion formula: In addition, the Plancherel formula holds: In both formulas is a linear representation of a group and The corollary above has an additional consequence: Lemma. Let be a group. Then the following is equivalent: is abelian. Every function on is a class function. All irreducible representations of have degree The induced representation As was shown in the section on properties of linear representations, we can - by restriction - obtain a representation of a subgroup starting from a representation of a group. Naturally we are interested in the reverse process: Is it possible to obtain the representation of a group starting from a representation of a subgroup? We will see that the induced representation defined below provides us with the necessary concept. Admittedly, this construction is not inverse but rather adjoint to the restriction. Definitions Let be a linear representation of Let be a subgroup and the restriction. Let be a subrepresentation of We write to denote this representation. Let The vector space depends only on the left coset of Let be a representative system of then is a subrepresentation of A representation of in is called induced by the representation of in if Here denotes a representative system of and for all and for all In other words: the representation is induced by if every can be written uniquely as where for every We denote the representation of which is induced by the representation of as or in short if there is no danger of confusion. The representation space itself is frequently used instead of the representation map, i.e. or if the representation is induced by Alternative description of the induced representation By using the group algebra we obtain an alternative description of the induced representation: Let be a group, a –module and a –submodule of corresponding to the subgroup of We say that is induced by if in