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Let $\Gamma\subset \mathbb{Q}^*$ be a finitely generated subgroup and let $p$ be a prime number such that the reduction group $\Gamma_p$ is a well defined subgroup of the multiplicative group $ \mathbb{F}^*_p$. In the first part of the talk, given that $\Gamma\subseteq \mathbb{Q}^*$ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the average over prime numbers, powers of the order of the reduction group $\Gamma_p$. The case of rank 1, when $\Gamma$ is generated by only one rational number, was previously considered by $C$. Pomerance and P. Kurlberg. In the second part of the talk, for any $m\in \mathbb{N}$ we determine an asymptotic formula for the average of the number of primes $p \leq x$ for which the index $[ \mathbb{F}^*_p:\Gamma_p]=m$. The average is performed over all finitely generated subgroups $\Gamma=\langle a_1, . . . , a_r\rangle\subset \mathbb{Q}$, with $a_i\in Z$ and $a_i\leq T_i$ with a range of uniformity: $T_i > exp(4(\log x \log \log x)^\frac{1}{2})$ for every $i = 1, . . . , r$. T he case of rank 1 and $m = 1$ corresponds to the classical Artin conjecture for primitive roots and has already been considered by P. J. Stephens in 1969. The second part of the talk is a joint work with Lorenzo Menici.