On the Order of Finitely Generated Subgroups of Q*(modp) and Divisors ofp−1

AbstractLetΓbe a finitely generated subgroup of Q* with rankr. We study the size of the order |Γp| ofΓmodpfor 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, yexplogτy] (τ∼0.15), we deduce that |Γp|⩾pr/(r+1)explogτpfor almost allpand, assuming the Generalized Riemann Hypothesis, we show that |Γp|⩾p/ψ(p) (ψ→∞) for almost allp. We also apply this to the Brown–Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots