Orders of simple groups and the Bateman--Horn Conjecture
Orders of simple groups and the Bateman--Horn Conjecture
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 Hölder in the 1890s.) The groups satisfying this condition are PSL2 (8), PSL2 (9) and PSL2 (p) for primes such that p2 − 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 Kn groups for each n ≥ 5.
Bateman-Horn Conjecture, Finite simple group, group order, prime degree, prime factor
257-269
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Zvonkin, Alexander K.
a3e21930-67d2-486f-90c3-0a4529352b35
September 2024
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Zvonkin, Alexander K.
a3e21930-67d2-486f-90c3-0a4529352b35
Jones, Gareth A. and Zvonkin, Alexander K.
(2024)
Orders of simple groups and the Bateman--Horn Conjecture.
International Journal of Group Theory, 13 (3), .
(doi:10.22108/ijgt.2023.136666.1828).
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 Hölder in the 1890s.) The groups satisfying this condition are PSL2 (8), PSL2 (9) and PSL2 (p) for primes such that p2 − 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 Kn groups for each n ≥ 5.
Text
2209.06510
- Accepted Manuscript
Text
IJGT_2024 Autumn_Vol 13_Issue 3_Pages 257-269
- Version of Record
More information
Accepted/In Press date: 6 May 2023
Published date: September 2024
Additional Information:
Funding Information:
The authors are grateful to Peter Cameron and Cheryl Praeger for helpful comments on applying the BHC to their work, and to Natalia Maslova for pointing out the connection with Kn groups. Alexander Zvonkin is supported by the French ANR project Combiné (ANR-19-CE48-0011).
Keywords:
Bateman-Horn Conjecture, Finite simple group, group order, prime degree, prime factor
Identifiers
Local EPrints ID: 486435
URI: http://eprints.soton.ac.uk/id/eprint/486435
ISSN: 2251-7650
PURE UUID: 286e49d2-15eb-4caa-a389-2c094bab4817
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Date deposited: 22 Jan 2024 17:37
Last modified: 17 Mar 2024 13:44
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Author:
Gareth A. Jones
Author:
Alexander K. Zvonkin
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