Groups of prime degree and the Bateman-Horn Conjecture

TL;DR

This paper explores the distribution of primes of a specific form related to finite simple groups and uses heuristic and computational methods based on the Bateman-Horn Conjecture to support a new conjecture about their infinitude.

Contribution

It proposes a conjecture that infinitely many primes of the form (q^n-1)/(q-1) exist for each prime n ≥ 3, supported by heuristic and computational evidence.

Findings

Heuristic arguments suggest the infinitude of such primes.

Computational evidence supports the conjecture.

Results extend to parameters of certain simple groups.

Abstract

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We present heuristic arguments and computational evidence based on the Bateman-Horn Conjecture to support a conjecture that for each prime $n\ge 3$ there are infinitely many primes of this form, even if one restricts to prime values of $q$. Similar arguments and results apply to the parameters of the simple groups ${\rm PSL}_n(q)$, ${\rm PSU}_n(q)$ and ${\rm PSp}_{2n}(q)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.