Abstract: In general,the mean-field-approximation(MFA) is adopted to solve the Gross-Pitaevskii equation in studies of superfluidity in Bose-Einstein condensates. However, we use the more rigorous exact diagonalizatio method(EDM) to investigate the state of a condensate with weak repulsive inter action in a two-dimensional N-boson system under rotation. The results show that the ground state is a fragmented condensate state instead of a single condensate state, contrary to what one might expect. By comparing the energy spectrm of the fragmented state in EDM with that given by the mean-field-approximation and calculating the conditional probability distribution, we find that the fragmented state is intrinsically unstable and tends to decay spontaneously to a single condensate state. The conditional probability distribution reveals the state after symmetry breaking, and vortex pictures similar to those under MFA are obtained. Moreover, on the basis of previous works on superfluidity which mainly considered the two extreme limits of weak interaction and strong Thomas-Fermi interaction we further study the ground-state properties of the system with medium interaction strength.