An asymptotic for the Hall–Paige conjecture

Sean Eberhard; Freddie Manners; Rudi Mrazović

doi:10.1016/j.aim.2022.108423

An asymptotic for the Hall–Paige conjecture

Sean Eberhard^*
, Freddie Manners^*
, Rudi Mrazović^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

8 Citations (Scopus)

Abstract

Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e^−1/2+o(1))|G^ab|n!²/nⁿ.

Original language	English
Article number	108423
Journal	Advances in Mathematics
Volume	404
Issue number	Part A
Early online date	28 Apr 2022
DOIs	https://doi.org/10.1016/j.aim.2022.108423
Publication status	Published - 06 Aug 2022
Externally published	Yes

Bibliographical note

Funding Information:
SE has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 803711 ); RM is supported in part by the Croatian Science Foundation under the project UIP-2017-05-4129 (MUNHANAP).

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

Hall–Paige conjecture
Latin squares
Transversals

ASJC Scopus subject areas

General Mathematics

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10.1016/j.aim.2022.108423Licence: Unspecified

Cite this

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abstract = "Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.",

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note = "Funding Information: SE has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 803711 ); RM is supported in part by the Croatian Science Foundation under the project UIP-2017-05-4129 (MUNHANAP). Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

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AU - Eberhard, Sean

AU - Manners, Freddie

AU - Mrazović, Rudi

N1 - Funding Information: SE has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 803711 ); RM is supported in part by the Croatian Science Foundation under the project UIP-2017-05-4129 (MUNHANAP). Publisher Copyright: © 2022 Elsevier Inc.

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N2 - Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.

AB - Hall and Paige conjectured in 1955 that a finite group G has a complete mapping if and only if its Sylow 2-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group G of order n satisfying the Hall–Paige condition is (e−1/2+o(1))|Gab|n!2/nn.

KW - Hall–Paige conjecture

KW - Latin squares

KW - Transversals

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DO - 10.1016/j.aim.2022.108423

M3 - Article

AN - SCOPUS:85129264749

SN - 0001-8708

VL - 404

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - Part A

M1 - 108423

ER -

An asymptotic for the Hall–Paige conjecture

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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