Permutation Representations
Associated to any finite set X is a vector space V = VX, which is the vector space with the elements of X as the basis (using any specified field). An action of a finite group G on X induces a linear action on V, called a permutation representation. The set of all finite dimensional representations of G has the structure of a ring, the representation ring, denoted R(G).
For a given G-set X, the character of the associated representation is
where <g> is the cyclic group generated by g.
The resulting map
taking a G-set to the corresponding representation is in general neither injective nor surjective.
The simplest example showing that β is not in general injective is for G = S3 (see table above), and is given by
Read more about this topic: Burnside Ring