Orders of simple groups and the Bateman--Horn Conjecture

Document Type : 2022 CCGTA IN SOUTH FLA

Authors

1 Department of Mathematics, School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK

2 LaBRI, Université de Bordeaux, 351 Cours de la Libération, F-33405, Talence, France

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 $p^2-1$ is a product of six primes. The conjecture suggests that there are infinitely many such primes $p$, 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, the construction of highly symmetric block designs, and the possible existence of infinitely many K$n$ groups for each $n\ge 5$.

Keywords

Main Subjects


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Volume 13, Issue 3 - Serial Number 3
2022 CCGTA IN SOUTH FLA
September 2024
Pages 257-269
  • Receive Date: 08 February 2023
  • Revise Date: 24 April 2023
  • Accept Date: 06 May 2023
  • Published Online: 01 September 2024