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We prove that a quantum spin chain with half-odd-integral spin cannot have a unique ground state with a gap, provided that the interaction is short ranged, translation invariant, and possesses time-reversal symmetry or ${\mathbb Z}_2 \times {\mathbb Z}_2$ symmetry (i.e., the symmetry with respect to the $\pi$ rotations of spins about the three orthogonal axes).
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