We consider the lattice Higgs model on $\mathbb{Z}^4$, with structure group
given by $ \mathbb{Z}_n $ for $ n \geq 2 $. We compute the expected value of
the Wilson loop observable to leading order when the gauge coupling constant
and hopping parameter are both sufficiently large. The leading order term is
expressed in terms of a quantity arising from the related but much simpler $
\mathbb{Z}_n $ model, which reduces to the Ising model when $n=2$. As part of
the proof, we construct a coupling between the lattice Higgs model and the $
\mathbb{Z}_n $ model.
