Groups of prime degree and the Bateman–Horn Conjecture

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 ${\mathrm{PSL}}_n\left(q\right)$ 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 ${\mathrm{PSL}}_n\left(q\right)$, ${\mathrm{PSU}}_n\left(q\right)$ and ${\mathrm{PSp}}_{2n}\left(q\right)$ which arise in the work of Dixon and Zalesskii on linear groups of prime degree.