Orders of simple groups and the Bateman--Horn Conjecture

Jones, Gareth Aneurin; Zvonkin, Alexander K.

doi:10.22108/ijgt.2023.136666.1828

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	Orders of simple groups and the Bateman--Horn Conjecture
International Journal of Group Theory
مقاله 6، دوره 13، شماره 3، آذر 2024، صفحه 257-269 اصل مقاله (251.55 K)
نوع مقاله: 2022 CCGTA IN SOUTH FLA
شناسه دیجیتال (DOI): 10.22108/ijgt.2023.136666.1828
نویسندگان
Gareth Aneurin Jones^* ¹؛ Alexander K. Zvonkin²
¹Department of Mathematics, School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK
²LaBRI, Université de Bordeaux, 351 Cours de la Libération, F-33405, Talence, France
چکیده
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 $n\ge 5$.
کلیدواژه‌ها
Finite simple group؛ group order؛ prime factor؛ prime degree؛ Bateman-Horn Conjecture

مراجع
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Orders of simple groups and the Bateman--Horn Conjecture