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Let (S, º) be a finite simple group,  s i  ,  i  = 1, ... ,  n , be fixed (not necessarily distinct) elements of  S , and let  E  α 1  ,  E  α 2  , ... ,  E  α  k +1   be a random realisation of a chain of states of a simple homogeneous irreducible Markov chain with the set of states { E  1 ,  E  2 , ... ,  E   n  }. We study convergence conditions and limit distributions for the sequences of random products of the form η ( k )  =  s  α 1   º  s  α 2   º ... º  s  α  k +1  . The convergence conditions are formulated in terms of some homomorphism from the fundamental group of the transition graph of the Markov chain to the structural group of the semigroup  S .

Connection between Markov chains on finite simple groups and fundamental groups