Orders of simple groups and the Bateman--Horn Conjecture

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Jones Gareth Aneurin; Zvonkin Alexander K.

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Journal Paper

Paper Information

Journal: INTERNATIONAL JOURNAL OF GROUP THEORY Year:2024 | Volume:13 | Issue:3 Page(s): 257-269

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Information Journal Paper

Title

Orders of simple groups and the Bateman--Horn Conjecture

Author(s)

Jones Gareth Aneurin | Zvonkin Alexander K. | Issue Writer Certificate

Keywords

finite simple group

group order

prime factor

prime degree

Bateman-Horn Conjecture

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 $n\ge 5$.

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