  In $SU(N)$ gauge theory, it is argued recently that there exists a "mixed
anomaly" between the CP symmetry and the 1-form $\mathbb{Z}_N$ symmetry at
$\theta=\pi$, and the anomaly matching requires CP to be spontaneously broken
at $\theta=\pi$ if the system is in the confining phase. In this paper, we
elaborate on this discussion by examining the large volume behavior of the
partition functions of the $SU(N)/\mathbb{Z}_N$ theory on $T^4$ a la 't Hooft.
The periodicity of the partition function in $\theta$, which is not $2\pi$ due
to fractional instanton numbers, suggests the presence of a phase transition at
$\theta=\pi$. We propose lattice simulations to study the distribution of the
instanton number in $SU(N)/\mathbb{Z}_N$ theories. A characteristic shape of
the distribution is predicted when the system is in the confining phase. The
measurements of the distribution may be useful in understanding the phase
structure of the theory.
