What's the procedure to condense several states into one in Markov chain? I am wondering the validity and the procedure to condense/group several states into one in markov chain, namely, if it is possible, how to transform the state vector and the transition matrix? Many thanks.

To generalize slightly the result mentioned by mjqxxxx, assume that one groups the original states $S_i$ into classes $C_a$. Hence the collection of classes $(C_a)$ is a partition of the state space and each class can be reduced to one state, or not. Then a sufficient condition for the resulting process to still be Markov is that the function $\varphi$ defined, for every state $S$ and class $C$, by the formula
$$
\varphi(S,C)=\sum_{T\in C}P(S\to T)
$$
depends on the state $S$ through its class only. In other words, one asks that, for every classes $C$ and $D$ and every states $S$ and $S'$ in $C$, 
$$
\varphi(S,D)=\varphi(S',D).
$$
Note that the condition is always true if every class is reduced to one state, and that it is also true if there is only one class.
Caveat: if this condition is not met, it can nevertheless happen that the resulting process is Markov for some, but not all, initial conditions.
