  Ground-state properties of a few spin-$1/2$ ultra-cold fermions confined in a
one-dimensional trap are studied by the exact diagonalization method. In
contrast to previous studies, it is not assumed that the projection of a spin
of individual particles is fixed. Therefore, the spin is treated as an
additional degree of freedom and the global magnetization of the system is
established spontaneously. Depending on the shape of the trap, inter-particle
interactions, and an external magnetic field, the phase diagram of the system
is determined. It is shown that, for particular confinements, some values of
the magnetization cannot be reached by the ground-state of the system.
