The summation of Markov sequences on a finite abelian group

We investigate the conditions under which the sum of independent Markov sequences on a finite abelian group $G$ is also a simple homogeneous Markov chain on the group $G$ with some matrix of transition probabilities. The considered problems concern the well-known procedure of consolidation of states of Markov chains. In this paper we develop a method based on the reduction of the initial problem to the solution of a system of special form of nonlinear equations over group algebras. We obtain new conditions under which sums of Markov chains on an arbitrary abelian group $G=Z_m$ are Markov chains, and necessary and sufficient conditions under which a sum of independent realisations of the initial Markov chains is also a simple homogeneous Markov chain.