  We study the antiferromagnetic spin exchange models with $S=1/2$ and S=1 on a
one-dimensional tetrahedron chain by both analytical and numerical approaches.
The system is shown to be effectively mapped to a decoupled spin chain in the
regime of strong rung coupling, and a spin sawtooth lattice in the regime of
weak rung coupling with spin $2S$ on the top row and spin $S$ on the lower row.
The ground state for the homogeneous tetrahedron chain is found to fall into in
the regime of strong rung coupling. As a result, the elementary excitation for
the spin-$ 1/2$ system is gapless whereas the excitation for the spin-1 system
has a finite spin gap. With the aid of the exact diagonalization method, we
determine the phase diagram numerically and find the existence of an additional
phase in the intermediate regime. This phase is doubly degenerate and is
characterized by an alternating distribution of rung singlet and rung spin
$2S$. We also show that the SU(3) exchange model on the same lattice has
completely different kind of ground state from that of its SU(2) correspondence
and calculate its ground state and elementary excitation analytically.
