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

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.
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: http://hdl.handle.net/11590/141125
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