The theory of finite groups can often be helpful in counting the number of certain configurations and in determining when two representations are equivalent or not. In this chapter, the authors introduce the concept of permutations as functions on a set of points, and also their representation in cycle notation. Permutation groups are the most important groups in the study of set systems. The basic operations that the people need to perform on permutations are multiplication, inversion, and conversion between cycle notation and array notation. It is useful to have programs that can switch between cycle notation and array notation. When searching for a particular set system, it is often advantageous to assume that the set system has certain automorphisms. It is significantly easier to develop an algorithm to compute the number of orbits of k-permutations than it is for k-subsets.