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We propose a simple model with the Z_N symmetry in order to answer whether the symmetry is a good concept in QCD with light quark mass. The model is constructed by imposing the flavor-dependent twisted boundary condition (TBC) on the three-flavor Polyakov-loop extended Nambu-Jona-Lasinio model. In the model, the Z_N symmetry is preserved below some temperature T_c, but spontaneously broken above T_c. Dynamics of the simple model is similar to that of the original PNJL model without the TBC, indicating that the Z_N symmetry is a good concept. We also investigate the interplay between the Z_N symmetry and the emergence of the quarkyonic phase.
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