Representation Theory by Ian Grojnowski.

Why Vector Spaces?

Definition: A representation of a group G on a vector space V is a homomorphism from G to GL(V). GL(V) is the set of linear maps from V to V.

If G is the dihedral group D_{8} acting on the set X of ordered pairs of vertices of the square, of which there are 16, then there are three orbits of G on X.