In this paper, we show that the set of all common fixed points of a one-parameter nonexpansive semigroup is that of some single nonexpansive mapping. We next compare our result with Bruck's famous fixed-point theorem. We finally prove very simple convergence theorems to a common fixed point. In our discussion, we assume neither the strict convexity of the underlying space, nor the weak compactness of the domain of a nonexpansive semigroup.