An asymptotic for the Hall-Paige conjecture

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)) vertical bar G(ab)vertical bar n!(2)/n(n). (c) 2022 Elsevier Inc. All rights reserved.