On the Order of Finitely Generated Subgroups of image*(mod p) and Divisors ofp−1 Original Research Article

LetΓbe a finitely generated subgroup of image* 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 primespless-than-or-equals, slantxfor whichp−1 has a divisor in an interval of the type [y, y exp logτ y] (τnot, vert, similar0.15), we deduce that Γpgreater-or-equal, slantedpr/(r+1) exp logτ pfor almost allpand, assuming the Generalized Riemann Hypothesis, we show that Γpgreater-or-equal, slantedp/ψ(p) (ψ→∞) for almost allp. We also apply this to the Brown–Zassenhaus Conjecture concerned with minimal sets of generators for primitive roots.