  In theories with discrete Abelian gauge groups, requiring that black holes be
able to lose their charge as they evaporate leads to an upper bound on the
product of a charged particle's mass and the cutoff scale above which the
effective description of the theory breaks down. This suggests that a
non-trivial version of the Weak Gravity Conjecture (WGC) may also apply to
gauge symmetries that are discrete, despite there being no associated massless
field, therefore pushing the conjecture beyond the slogan that `gravity is the
weakest force'. Here, we take a step towards making this expectation more
precise by studying $\mathbb{Z}_N$ and $\mathbb{Z}_2^N$ gauge symmetries
realised via theories of spontaneous symmetry breaking. We show that applying
the WGC to a dual description of an Abelian Higgs model leads to constraints
that allow us to saturate but not violate existing bounds on discrete
symmetries based on black hole arguments. In this setting, considering the
effect of discrete hair on black holes naturally identifies the cutoff of the
effective theory with the scale of spontaneous symmetry breaking, and provides
a mechanism through which discrete hair can be lost without modifying the
gravitational sector. We explore the possible implications of these arguments
for understanding the smallness of the weak scale compared to $M_{Pl}$.
