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Given a finitely generated multiplicative subgroup of rational numbers$\Gamma$, assuming the Generalized Riemann Hypothesis, we determine anasymptotic formula for average over prime numbers, powers of the order of thereduction group modulo $p$. The problem was considered in the case of rank $1$by Pomerance and Kurlberg. In the case when $\Gamma$ contains only positivenumbers, we give an explicit expression for the involved density in terms of anEuler product. We conclude with some numerical computations.

Average Multiplicative Order of Finitely Generated Subgroup of Rational Numbers Over Primes