We have already made extensive use of stochastic matrices in developing the results in Chapters 2,3, and 4. In this chapter, we specifically focus on stochastic matrices and their associated group inverses in the theory of discrete-time, time homogeneous Markov chains on a finite state space. We begin with a brief introduction to Markov chains in section 5.1, and then consider the special case of periodic stochastic matrices in section 5.2. This is followed in section 5.3 by an analysis of the conditioning properties of the stationary distribution under perturbation of the underlying stochastic matrix, and the use of the group inverse in deriving bounds on the subdominant eigenvalues of a stochastic matrix in section 5.4. The chapter concludes with a few illustrative examples in section 5.5.