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PRIMITIVE PRIME DIVISORS AND THE $\mathbf{n}$TH CYCLOTOMIC POLYNOMIAL

Part of: Linear algebraic groups and related topics Computational number theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  04 November 2015

S. P. GLASBY ,
FRANK LÜBECK ,
ALICE C. NIEMEYER  and
CHERYL E. PRAEGER
S. P. GLASBY
Affiliation:
Centre for Mathematics of Symmetry and Computation, University of Western Australia, Australia The Department of Mathematics, University of Canberra, Australia email GlasbyS@gmail.com
FRANK LÜBECK
Affiliation:
Lehrstuhl D für Mathematik, RWTH Aachen University, Pontdriesch 14/16, 52062 Aachen, Germany email Frank.Luebeck@Math.RWTH-Aachen.De
ALICE C. NIEMEYER*
Affiliation:
Lehr- und Forschungsgebiet Algebra, RWTH Aachen University, Pontdriesch 10-16, 52062 Aachen, Germany email Alice.Niemeyer@math.rwth-aachen.de
CHERYL E. PRAEGER
Affiliation:
Centre for Mathematics of Symmetry and Computation, University of Western Australia, Australia King Abdulaziz University, Jeddah, Saudi Arabia email Cheryl.Praeger@uwa.edu.au
*
✉Alice.Niemeyer@nuim.ie
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Abstract

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Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$, which is closely related to the cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{n}(x)$ and to primitive prime divisors of $q^{n}-1$. Our definition of $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we provide an algorithm for determining all pairs $(n,q)$ with $\unicode[STIX]{x1D6F7}_{n}^{\ast }(q)\leq cn^{k}$. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Keywords

Zsigmondy primescyclotomic polynomial

MSC classification

Primary: 11T22: Cyclotomy
Secondary: 11Y40: Algebraic number theory computations 20G05: Representation theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 1 , February 2017 , pp. 122 - 135
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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