An asymptotic for the Hall--Paige conjecture

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 $(e^{-1/2}+o(1))|G^{\text{ab}}|n{!}^2/n^n$.

58 pages, substantial changes to v1 following referee comments. To appear in Advances in Mathematics

References in corpus (1)

• The Hall--Paige conjecture, and synchronization for affine and diagonal groups

Cited by in corpus (2)

• Transversals in quasirandom latin squares
• Degree of Orthomorphism Polynomials over Finite Fields