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 (qn−1)/(q−1) of PSLn(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≥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 PSLn(q), PSUn(q) and PSp2n(q) which arise in the work of Dixon and Zalesskii on linear groups of prime degree.