Cluster synchronization is a phenomenon in which a network self-organizes into a pattern of synchronized sets. It has been shown that diverse patterns of stable cluster synchronization can be captured by symmetries of the network. Here we establish a theoretical basis to divide an arbitrary pattern of symmetry clusters into independently synchronizable cluster sets, in which the synchronization stability of the individual clusters in each set is decoupled from that in all the other sets. Using this framework, we suggest a new approach to find permanently stable chimera states by capturing two or more symmetry clusters---at least one stable and one unstable---that compose the entire fully symmetric network.
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