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This paper investigates the group synchronization problem for interacting clusters of agents, which can be either generic linear systems or nonlinear oscillators, by focusing on in-depth understanding of how the couplings among agents influence the group behavior. To this end, we work on both the homogeneous case that agents have the same system dynamics and the heterogeneous case that agents from different clusters have different system dynamics. It is shown that the synchronization for each cluster of agents is irrelevant to the strength of the intercluster couplings as long as the intercluster couplings are collectively acyclic; that is, the intercluster couplings do not incur desynchronization and, thus, no extra effort beyond the strengths of the intracluster couplings, which are necessary to guarantee the synchronization of each cluster in the absence of intercluster couplings, are needed. Moreover, we provide …