Reductions of subgroups of the multiplicative group

Let $K$ be a number field, and let $G\subset K^\times$ be a finitely generated subgroup. Fix some prime number $\ell$, and consider the set of primes $\mathfrak{p}$ of $K$ satisfying the following property: the reduction of $G$ modulo $\mathfrak{p}$ is well-defined and has size coprime to $\ell$. We give a closed--form expression for the density of this set.

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