In Chap. 1 it is explained that if p is a probability on a finite group G the group matrix XG(p) is a transition matrix for a random walk on G. If f is an arbitrary function on G the process of transforming XG(f) into a block diagonal matrix is equivalent to the obtaining the Fourier transform of f. This chapter explains the connections with harmonic analysis and the group matrix. Most of the discussion is on probability theory and random walks. The fusion of characters discussed in Chap. 4 becomes relevant, and also the idea of fission of characters is introduced, especially those fissions which preserve diaonalizability of the corresponding group matrix. As an example of how the group matrix and group determinant can be used as tools, their application to random walks which become uniform after a finite number of steps is examined.