Period of a state in a Markov chain ($d(i)$)

The period of a state $i$ is the greatest common divisor of its positive-probability return times:A state is aperiodic when its period is one.

If communicating states $i,j$ can be joined in $r$ and $s$ steps, then $r+s$ is a return time to $i$, and adjoining any return loop of length $n$ at $j$ makes $r+n+s$ another. Their difference shows $d(i)\mid n$, hence $d(i)\mid d(j)$; symmetry gives equality.