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 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 ≥ 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 PSL n (q), PSU n (q) and PSp 2n (q) which arise in the work of Dixon and Zalesskii on linear groups of prime degree.