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The central theorem in the finite group case is the Frobenius reciprocity theorem. It is stated in terms of another construction of representations, the restriction map (which is a functor): any linear representation of G, as K[G]- module where K[G] is the group ring of G over a field K, is also a K[H]-module. The theorem states that, given representations of G and σ of H, the space of G- intertwining maps from ρ to Ind(σ) has the same dimension as that of the H-intertwining maps from Res(ρ) to σ. (Here Res stands for restricted representation, and Ind for induced representation.) It is useful (in the typical case of non-modular representations, anyway - say with K = C) for computing the decomposition of the induced representation: we can do calculations on the side of H, which is the 'small' group.

In fact, anachronistically, we can recognise that this theorem shows that Res and Ind are adjoint functors. The content of that statement is more than the dimensions: it requires that the isomorphism of vector spaces of intertwining maps be natural, in the sense of category theory. It actually suggests that induced representation can in this case be defined by means of the adjunction. That's not the only way to do it - and perhaps not the only helpful way - but it means that the theory will not be ad hoc in its start.

One can therefore makes the reciprocity theorem the way to defining the induced representation. There is another way, suggested by the permutation examples of the introductory paragraph. The induced representation Ind(σ) should be realized as a space of functions on G transforming under H according to the representation σ. Therefore if σ acts on the vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for V, we should look at V-valued functions on G on which H acts via σ (this must be said carefully with explicit talk about left- and right-actions). This approach allows the induced representation to be a kind of free moduleIn mathematics, a free module is a module having a free basis''. For an R module M the set E e e . e is a free basis for M if and only if: 1) E is a generating set for M that is to say every element of M is a sum of elements of E multiplied by coefficient construction.

The two approaches outlined above can be reconciled in the case of finite groups, by using the tensor productAbstract algebra Algebra In mathematics, the tensor product denoted by , may be applied in different contexts to vectors, matrices, tensors and vector spaces. In each case the significance of the symbol is the same: the most general bilinear operation. with K[G] as a K[H]-module. There is a third and classical approach, of simply writing down the character (trace) of the induced representation, in terms of conjugation in G of elements g into H.

In more general terms, the reciprocity theorem isn't available in generality for representations of topological groupIn mathematics, a topological group ''G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. Here, G × G is viewed as a topological space by using the produs; and the character formulas are also subject to some analytical problems. The second definition, on the other hand, is a major theme in harmonic analysisHarmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms. The basic waves are called ", in generality. It is adapted to the theory of vector bundleIn mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topolos, for example.