  We propose the definitions of many-body topological invariants to detect
symmetry-protected topological phases protected by point group symmetry, using
partial point group transformations on a given short-range entangled quantum
ground state. Partial point group transformations $g_D$ are defined by point
group transformations restricted to a spatial subregion $D$, which is closed
under the point group transformations and sufficiently larger than the bulk
correlation length $\xi$. By analytical and numerical calculations,we find that
the ground state expectation value of the partial point group transformations
behaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i \theta+
\gamma - \alpha \frac{{\rm Area}(\partial D)}{\xi^{d-1}} \Big]$. Here, ${\rm
Area}(\partial D)$ is the area of the boundary of the subregion $D$, and
$\alpha$ is a dimensionless constant. The complex phase of the expectation
value $\theta$ is quantized and serves as the topological invariant, and
$\gamma$ is a scale-independent topological contribution to the amplitude. The
examples we consider include the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$
invariants of topological superconductors protected by inversion symmetry in
$(1+1)$ and $(3+1)$ dimensions, respectively, and the lens space topological
invariants in $(2+1)$-dimensional fermionic topological phases. Connections to
topological quantum field theories and cobordism classification of
symmetry-protected topological phases are discussed.
