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

Published online by Cambridge University Press:  04 November 2015


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

Abstract

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.


Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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References

Bamberg, J. and Penttila, T., ‘Overgroups of cyclic sylow subgroups of linear groups’, Comm. Algebra 36 (2008), 25032543.CrossRefGoogle Scholar
Bang, A. S., ‘Taltheoretiske Undersøgelser’, Tidsskr. Math. (5) 4 (1886), 7080; and 130–137.Google Scholar
Biliotti, M., Jha, V., Johnson, N. L. and Montinaro, A., ‘Translation planes of order q 2 admitting a two-transitive orbit of length q + 1 on the line at infinity’, Des. Codes Cryptogr. 44 (2007), 6986.CrossRefGoogle Scholar
Biliotti, M., Jha, V., Johnson, N. L. and Montinaro, A., ‘Two-transitive groups on a hyperbolic unital’, J. Combin. Theory Ser. A 115 (2008), 526533.CrossRefGoogle Scholar
Biliotti, M., Jha, V., Johnson, N. L. and Montinaro, A., ‘Classification of projective translation planes of order q 2 admitting a two-transitive orbit of length q + 1’, J. Geom. 90(1–2) (2008), 100140.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24(3–4) (1997), 235265.CrossRefGoogle Scholar
Camina, A. R. and Whelan, E. A., Linear Groups and Permutations, Research Notes in Mathematics, 118 (Pitman (Advanced Publishing Program), Boston, MA, 1985).Google Scholar
Dandapat, G. G., Hunsucker, J. L. and Pomerance, C., ‘Some new results on odd perfect numbers’, Pacific J. Math. 57 (1975), 359364.CrossRefGoogle Scholar
DiMuro, J., ‘On prime power order elements of general linear groups’, J. Algebra 367 (2012), 222236.CrossRefGoogle Scholar
Dummit, D. S. and Foote, R. M., Abstract Algebra, 3rd edn, (Wiley & Sons, Hoboken, 2004).Google Scholar
Feit, W., ‘On large Zsigmondy primes’, Proc. Amer. Math. Soc. 102 (1988), 2936.CrossRefGoogle Scholar
The GAP Group, GAP – Groups, Algorithms, and Programming, version 4.7.7; 2015,http://www.gap-system.org.Google Scholar
Glasby, S. P., Supporting GAP and Magma code, http://www.maths.uwa.edu.au/∼glasby/RESEARCH.Google Scholar
Guralnick, R., Penttila, T., Praeger, C. E. and Saxl, J., ‘Linear groups with orders having certain large prime divisors’, Proc. Lond. Math. Soc. 78 (1999), 167214.CrossRefGoogle Scholar
Hering, C., ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order’, Geom. Dedicata 2 (1974), 425460.CrossRefGoogle Scholar
James, G. D., ‘On the minimal dimensions of irreducible representations of symmetric groups’, Math. Proc. Cambridge Philos. Soc. 94 (1983), 417424.CrossRefGoogle Scholar
Lenstra–Pomerance–Wagstaff conjecture, http://primes.utm.edu/mersenne/heuristic.html.Google Scholar
Lüneburg, H., ‘Ein einfacher Beweis für den Satz von Zsigmondy über primitive Primteiler von A N - 1. [A simple proof of Zsigmondy’s theorem on primitive prime divisors of A N - 1]’, in: Geometries and Groups (Springer, Berlin, 1981), 219222.CrossRefGoogle Scholar
Niemeyer, A. C. and Praeger, C. E., ‘A recognition algorithm for classical groups over finite fields’, Proc. Lond. Math. Soc. 77 (1998), 117169.CrossRefGoogle Scholar
Ribenboim, P., The Little Book of Big Primes (Springer, New York, 1991).CrossRefGoogle Scholar
Roitman, M., ‘On Zsigmondy primes’, Proc. Amer. Math. Soc. 125 (1997), 19131919.CrossRefGoogle Scholar
Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. 3 (1892), 265284.CrossRefGoogle Scholar

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