Primitive prime divisors and the $n$-th cyclotomic polynomial

TL;DR

This paper introduces a new number theoretic quantity related to cyclotomic polynomials and primitive prime divisors, providing an algorithm to classify certain subgroup families of finite linear groups.

Contribution

It defines a novel version of *_n(q) and proves its equivalence to Hering's definition, along with an algorithm for classifying pairs (n,q) based on this quantity.

Findings

Extended and corrected Hering's subgroup classification results.

Provided an algorithm to determine pairs (n,q) with *_n(q)  c n^k.

Established the equivalence of the new *_n(q) with Hering's original definition.

Abstract

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