I am going through Knowing the Odds by John B. Walsh, and I am stuck at one of the exercises there (which is important to understand some next theorem).
The exercise is as follows:
Let $X$ be an irreducible chain with period $d$. Show that there exist $d$ disjoint sets of states, $S_1,\dots,S_d$, such that if the chain starts in $S_1$, it rotates through $S_2, S_3, \dots ,S_d$ and then back to $S_1$.
I don't even understand what he exactly means by "rotation" and do not know where to start. Any thoughts? Where do I start the proof?