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An asymptotic for the Hall--Paige conjecture
- Publication Year :
- 2020
-
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 $(e^{-1/2} + o(1)) \, |G^\text{ab}| \, n!^2/n^n$.<br />Comment: 58 pages, substantial changes to v1 following referee comments. To appear in Advances in Mathematics
- Subjects :
- Mathematics - Combinatorics
Mathematics - Group Theory
05B15
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2003.01798
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.aim.2022.108423