Orders of simple groups and the Bateman--Horn Conjecture

TL;DR

This paper investigates the potential infinitude of simple groups with orders as products of six primes, using the Bateman--Horn Conjecture to provide heuristic evidence and explore implications in group theory.

Contribution

It applies the Bateman--Horn Conjecture to predict the existence of infinitely many simple groups with specific prime factorization properties, linking number theory and group classification.

Findings

Heuristic estimates suggest infinitely many primes p with p^2-1 having six prime factors.

Computational evidence supports the conjecture's predictions.

Implications for classification of permutation and linear groups, and for combinatorial designs.

Abstract

We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and H\"older in the 1890s.) The groups satisfying this condition are ${\rm PSL}_2(8)$, ${\rm PSL}_2(9)$ and ${\rm PSL}_2(p)$ for primes $p$ such that $p^2-1$ has just six prime factors. The conjecture suggests that there are infinitely many such primes, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, and the construction of highly symmetric block designs.