 Solution

True. For a state $i$, letbe its possible return times under $P$, and define ${\mathcal{R}}_Q(i)$ similarly. Every positive $P$-path is a positive $Q$-path, soThe greatest common divisor of the larger set divides that of the smaller set. Since the latter is one by the assumed aperiodic Markov chain property, the former is also one. Hence every state is aperiodic under $Q$.