Group Actions

In the previous chapter, we looked at how the symmetry group of a regular polygon permutes its vertices, and how the symmetry group of a cube permutes the four diagonals. Realizing these groups as permutation groups of a set of objects told us a lot about them. In this chapter, we are going to pursue this point of view. In general, one says that a group G acts on a set X (usually finite) if one is given a homomorphism G → S_X . A more convenient way to express this is the following.