On the order of finitely generated subgroups of Q* and divisors of p-1

LetΓbe a finitely generated subgroup of View the MathML source* with rankr. We study the size of the order |Γp| ofΓ mod pfor density-one sets of primes. Using a result on the scarcity of primesp⩽xfor whichp−1 has a divisor in an interval of the type [y, y exp logτ y] (τ∼0.15), we deduce that |Γp|⩾pr/(r+1) exp logτ pfor almost allpand, assuming the Generalized Riemann Hypothesis, we show that |Γp|⩾p/ψ(p) (ψ→∞) for almost all p. We also apply this to the Brown–Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots.