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.

Pappalardi, F. (1996). On the order of finitely generated subgroups of Q* and divisors of p-1. JOURNAL OF NUMBER THEORY, 57(2), 207-222 [10.1006/jnth.1996.0044,].

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

PAPPALARDI, FRANCESCO
1996-01-01

Abstract

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.
1996
Pappalardi, F. (1996). On the order of finitely generated subgroups of Q* and divisors of p-1. JOURNAL OF NUMBER THEORY, 57(2), 207-222 [10.1006/jnth.1996.0044,].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/141125
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