It is an amazing fact that remarkably complex behaviors could emerge from a large collection of very rudimentary dynamical agents through very simple local interactions. However, it still remains elusive on how to design these local interactions among agents so as to achieve certain desired collective behaviors. This paper aims to tackle this challenge and proposes a divide-and-conquer approach to guarantee specified global behaviors through local coordination and control design for multi-agent systems. The basic idea is to decompose the requested global specification into subtasks for each individual agent. It should be noted that the decomposition is not arbitrary. The global specification should be decomposed in such a way that the fulfilment of these subtasks by each individual agent will imply the satisfaction of the global specification as a team. First, it is shown by a counterexample that not all specifications can be decomposed in this sense. Then, a natural follow-up question is what the necessary and sufficient condition should be for the proposed decomposability of a global specification. The main part of the paper is set to answer this question. The case of two cooperative agents is investigated first, and a necessary and sufficient condition is presented and proven. Later on, the result is generalized to the case of arbitrary finite number of agents, and a hierarchical algorithm is proposed, which is shown to be a sufficient condition. Finally, a cooperative control scenario for a team of three robots is developed to illustrate the task decomposition procedure.