Is this statement about periodic communicating classes an equivalence statement?

Question

In our reading about Markov chains we had the following theorem (including proof):

Let $(X_n)_{n\in\mathbb{N}_0}$ denote a Markov chain with state space $E$. A periodic communicating class $C\subseteq E$ with period $p$ can be decomposed into a disjoint union of sets $C_0\cup C_1\ldots\cup\ldots\cup C_{p-1}$ in such a way that, if $i\in C_n$ and$j\in C_m$ with $m\neq n+1\text{ mod }p$, then $p_{ij}=0$.

Now it is asked if this statement is an equivalence statement.

What would be the statement that would make the theorem an equivalence statement? I cannot see it clear. In other words: What do I have to prove or disprove?

Answer 1

The missing part of the equivalence statement would be:

If $p_{ij}=0$ then $i\in C_n$ and $j\in C_m$ with $m\neq n+1\text{ mod }p$.

This is wrong in general, consider the Markov chain on $\{0,1,2,3,4\}$ with $p_{01}=p_{02}=\frac12$ and $p_{13}=p_{24}=p_{30}=p_{40}=1$, for $i=2$ and $j=3$.

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Edit: A rough sketch of a Markov chain with period $3$:

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