Counter example on the period of Markov Chain

Question

I want to find a counter example for the following statement:

"If an irreducible Markov chain has period 2, then for every state $i \in S$ we have $P_{ii}^2 > 0$"

I feel like that we can't really find a counterexample for this statement, since $2$ is the gcd of itself.

Do you guys have any thoughts?

Answer 1

Consider $P=\begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix}$.