  We investigate a three-site ring system with a small number of quantum
degenerate bosons and fermions. By means of the exact diagonalization of the
Bose-Fermi-Hubbard Hamiltonian, we show that the symmetry of the ground state
configuration is a function of both the boson-boson and the inter-species
interaction in the system. The phase diagram of the system, constructed by
computing the exact two-body spatial correlations, reveals nontrivial
insulating phases that exist even in the strong bosonic tunneling limit and for
incommensurate filling of bosons. These insulating phases are due to the
inter-species interactions in the system and are not necessarily accompanied by
the suppression of the particle number fluctuations.
