  We explore constraints on (1+1)$d$ unitary conformal field theory with an
internal $\mathbb{Z}_N$ global symmetry, by bounding the lightest
symmetry-preserving scalar primary operator using the modular bootstrap. Among
the other constraints we have found, we prove the existence of a
$\mathbb{Z}_N$-symmetric relevant/marginal operator if $N-1 \le c\le 9-N$ for
$N\leq4$, with the endpoints saturated by various WZW models that can be
embedded into $(\mathfrak{e}_8)_1$. Its existence implies that robust gapless
fixed points are not possible in this range of $c$ if only a $\mathbb{Z}_N$
symmetry is imposed microscopically. We also obtain stronger, more refined
bounds that depend on the 't Hooft anomaly of the $\mathbb{Z}_N$ symmetry.
