Using the center manifold theory for maps in a systematic way, we present in this note a necessary condition and sufficient conditions for discrete-time nonlinear systems to be stabilizable via smooth state feedback. Then, we derive a necessary and sufficient condition characterizing the existence of exponential observers for Lyapunov stable discrete-time nonlinear systems. This, in turn, leads to a separation principle for the asymptotic stabilization problem. As an application, the output regulation problem is solved via error feedback.