Suppose that $X_1,X_2,\ldots$ is a time-homogeneous irreducible and aperiodic Markov chain over a finite state space $\mathcal{X}$.
Now consider the process $Y_1,Y_2,\ldots$ over the state space $\mathcal{X}^m$ of blocks obtained by grouping together $m$ symbols of $X_j$'s, i.e., $$Y_i = (X_{(i-1)m+1},X_{(i-1)m+2},\ldots,X_{im}).$$ Is $Y_1,Y_2,\ldots$ also an irreducible and aperiodic Markov chain?