where is a complex vector space. The representation is called

In general, linear maps that satisfy this equation are called -maps or intertwiners. The investigation of representations of general finite groups was soon picked up by Burnside, Schur, and many others. The main results of their work can be formulated as follows:

- (1)
- Every representation of a finite group is unitarizable. This means that there exists a Hilbert space structure on such that .
- (2)
- Consequently, every representation of is equivalent to a direct sum of irreducible representations.
- (3)
- (Schur's lemma) When and are irreducible and inequivalent, the only intertwiner between and is . On the other hand, when and are equivalent, every nonzero intertwiner is an isomorphism. In particular, when every intertwiner is scalar.
- (4)
- By (3), the decomposition in (2) is essentially unique.
- (5)
- The left regular representation of is defined by the
left multiplication of on itself. More precisely, we take
as the space of complex valued functions on , and define the
left regular representation by
.
Similarly, we define the right regular representation
on by the right multiplication of on itself.
Then is a representation via
.
As such, is equivalent to the direct sum

where runs over the set of all equivalence classes of irreducible representations of . - (6)
- (Plancherel formula) If , we define
its Fourier transform as the function on
such that
.
We give a Hilbert space structure by the Hermitian
inner product
. Then

Here denotes the Hilbert-Schmidt norm of .