  In this letter we consider a magnetic impurity in a one-dimensional
spin-$1/2$ Fermi gas with infinitely strong repulsive interaction between
fermions. We rigorously prove that, independent of whether the magnetic
coupling between impurity and fermions is ferromagnetic or anti-ferromagnetic,
the ground state is always a fully polarized ferromagnetic state for the
itinerant fermions. This ferromagnetism can be understood as a cooperative
effect of avoiding frustration of magnetic coupling during fermion hopping and
the large spin degeneracy of a fermion Tonks gas. By numerically diagonalizing
a finite size system, we show that the spin gap first increases linearly with
magnetic coupling strength in the weak coupling regime, while decreases in the
strong coupling regime. Our results show that a magnetic impurity in a strongly
correlated gas can exhibit effect different from the Kondo effect as in a
weakly correlated Fermi liquid.
