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