Let {X n : n = 0, 1,...} be a Markov chain with state space S and probability transition matrix P = (p ij ). Show that periodicity is a class property. That is, for i, j ∈ S, if i and j communicate...

Let {X_n
: n = 0, 1,...} be a Markov chain with state space S and probability transition matrix P = (p_ij
). Show that periodicity is a class property. That is, for i, j ∈ S, if i and j communicate with each other, then they have the same period.