I am reading through Introduction to stochastic models by Roe Goodman, but I am a little confused by an answer to one of the exercises.
Consider the below Markov chain. In my mind the class {1,3} is aperiodic, since for example we could take the path $1 \to 1$ which has length 1, or the path $1 \to 3 \to 1$, which has length 3. Clearly since 3 and 1 are only divisible by 1, the state 1 is aperiodic. We can then make a similar argument for the state 3, and thus the class is aperiodic.
A solutions manual I am using claims that this is a periodic class with period 2 however. Have I made a mistake?
