= Solution
For a unique gapped ground state, one-dimensional bosonic phases protected by an on-site symmetry $G$ are classified by the cohomology class
$$
\boxed{[\omega]\in H^2(G,U(1))}
$$
of a <projective virtual symmetry of a matrix product state>. After blocking finitely many sites, an <injective matrix product state> tensor satisfies
$$
\sum_jU(g)_{ij}A^j=e^{i\theta(g)}V(g)A^iV(g)^{-1},
\qquad
V(g)V(h)=\omega(g,h)V(gh).
$$
Changing the virtual gauge or rephasing $V(g)$ changes $\omega$ by a coboundary and leaves $[\omega]$ fixed.
If symmetry breaking and degenerate ground states are allowed, the additional data are the unbroken subgroup $H\subseteq G$, the way $G$ permutes the broken-symmetry sectors, and a class in $H^2(H,U(1))$ within one sector. Acting with $\bigotimes_iU_i(g)$ preserves a symmetric ground state up to phase in the unbroken case; in a broken phase it generally permutes the different ground states.
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