Transition semigroup of a continuous-time Markov chain ($P(t)$)

For a continuous-time Markov chain, $P_{ij}(t)={\mathbb{P}}_i(X_t=j)$ and the matrices satisfy $P(s+t)=P(s)P(t)$. On a finite state space with Q-matrix $Q$, $P(t)=e^{tQ}$.