We study hidden-charm pentaquarks in the baryon-meson molecule picture within the framework of the constituent quark model, considering all the possible quark configurations for the ground state. The strong decays of the Pc(4312), Pc(4440), and Pc(4457) resonances are investigated. It is found that pentaquark states of all quark configurations in the model can decay through the N J/ψ channel, but only five pentaquark states may decay dominantly in open-charm modes. The partial decay widths in the pηc channel are in the same order as in the pJ/ψ channel for J P = 1/2 − baryonmeson pentaquark states, but the pηc channel is forbidden for J P = 3/2 − states. For Pc(4440) and Pc( 4457), the pηc, Σc D and Λ + c D channels are open only for J P = 1/2 − and the Σ * c D channel is open only for J P = 3/2 − while for Pc(4312), due to the mass threshold of Σc D and Σ * c D higher than 4312 MeV, the pηc and Λ + c D channels are open for J P = 1/2 − but no open-charm cannel open for J P = 3/2 − . We strongly suggest that the spin of the Pc resonances may be determined in experiments by investigating the pηc and open-charm channels.

Pc resonances are studied in the approach of quark model and group theory. It is found that there are totally 17 possible pentaquark states with the quark contents $q^3Q \bar{Q}$ (q are u and d quarks; Q is c quark) in the compact pentaquark picture, where the hidden heavy pentaquark states may take the color singlet-singlet ($[111]_{{qqq}}\otimes [111]_{{c \bar c}}$) and color octet-octet ($[21]_{{qqq}}\otimes [21]_{{c \bar c}}$) configurations. The partial decay widths of hidden heavy pentaquark states are calculated for all possible decay channels. The results show that the p J/φ is the dominant decay channel for both the spin 3/2 and 1/2 pentaquark states, and indicate that the Pc(4440) may not be a compact pentaquark state while Pc(4312) and Pc(4457) could be the spin-$\frac{1}{2}$ and spin-$\frac{3}{2}$ pentaquark states, respectively.