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  • S. P. GLASBY (a1) (a2), FRANK LÜBECK (a3), ALICE C. NIEMEYER (a4) and CHERYL E. PRAEGER (a1) (a5)

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.

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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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