TY - JOUR
ID - 27521
TI - Orders of simple groups and the Bateman--Horn Conjecture
JO - International Journal of Group Theory
JA - IJGT
LA - en
SN - 2251-7650
AU - Jones, Gareth Aneurin
AU - Zvonkin, Alexander K.
AD - Department of Mathematics, School of Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, UK
AD - LaBRI, UniversiteĢ de Bordeaux, 351 Cours de la LibeĢration, F-33405, Talence, France
Y1 - 2024
PY - 2024
VL - 13
IS - 3
SP - 257
EP - 269
KW - finite simple group
KW - group order
KW - prime factor
KW - prime degree
KW - Bateman-Horn Conjecture
DO - 10.22108/ijgt.2023.136666.1828
N2 - 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 $n\ge 5$.
UR - https://ijgt.ui.ac.ir/article_27521.html
L1 - https://ijgt.ui.ac.ir/article_27521_b123f0c513f61f9c023d31a0e1bfa7c8.pdf
ER -