The effective Hamiltonian, obtained from the Hubbard model in the strong-coupling limit, is diagonalized exactly for a periodic two-dimensional square lattice of ten sites. We obtain the ground state of the system for any filling of the lattice and the K-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio tU, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing tU, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of tU exists for which pairing of the holes may be possible. © 1988 The American Physical Society.