  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.
