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We study $S=1$ quantum spin systems on the infinite chain with short rangedHamiltonians which have certain rotational and discrete symmetry. We define a$\mathbb{Z}_2$ index for any gapped unique ground state, and prove that it isinvariant under smooth deformation. By using the index, we provide the firstrigorous proof of the existence of a "topological" phase transition, whichcannot be characterized by any conventional order parameters, between the AKLTground state and trivial ground states. This rigorously establishes that theAKLT model is in a nontrivial symmetry protected topological phase.

Topological phase transition and $\mathbb{Z}_2$ index for $S=1$ quantum spin chains