  Recently, it was realized that quantum states of matter can be classified as
long-range entangled (LRE) states (i.e. with non-trivial topological order) and
short-range entangled (SRE) states (\ie with trivial topological order). We can
use group cohomology class ${\cal H}^d(SG,R/Z)$ to systematically describe the
SRE states with a symmetry $SG$ [referred as symmetry-protected trivial (SPT)
or symmetry-protected topological (SPT) states] in $d$-dimensional space-time.
In this paper, we study the physical properties of those SPT states, such as
the fractionalization of the quantum numbers of the global symmetry on some
designed point defects, and the appearance of fractionalized SPT states on some
designed defect lines/membranes. Those physical properties are SPT invariants
of the SPT states which allow us to experimentally or numerically detect those
SPT states, i.e. to measure the elements in ${\cal H}^d(G, R/Z)$ that label
different SPT states. For example, 2+1D bosonic SPT states with $Z_n$ symmetry
are classified by a $Z_n$ integer $m \in {\cal H}^3(Z_n, R/Z)=Z_n$. We find
that $n$ identical monodromy defects, in a $Z_n$ SPT state labeled by $m$,
carry a total $Z_n$-charge $2m$ (which is not a multiple of $n$ in general).
