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.

由于有限单群的分类，素数置换群的分类是完备的，除了自然次数什么时候出现的问题$(q^n-\mathrm{1个})/(q-\mathrm{1个})$的${\mathrm{PSL}}_n\left(q\right)$是质数。我们提出了基于贝特曼-霍恩猜想的启发式论证和计算证据，以支持对每个素数的猜想$n\ge \mathrm{3个}$有无限多个这种形式的素数，即使一个人限制为素数$q$. 类似的论点和结果适用于简单群的参数${\mathrm{PSL}}_n\left(q\right)$,${\mathrm{电源}}_n\left(q\right)$和${\mathrm{聚苯乙烯}}_{\mathrm{2个}n}\left(q\right)$它出现在 Dixon 和 Zalesskii 关于素数线性群的工作中。