论文标题
Hall的渐近造型-Paige猜想
An asymptotic for the Hall--Paige conjecture
论文作者
论文摘要
Hall and Paige在1955年猜想，有限的组$ G $在且仅当其Sylow $ 2 $ -SUBGROUPS是琐碎的或非循环的时才具有完整的映射。 Wilcox，Evans和Bray在2009年使用有限的简单组和广泛的计算机代数证明了这一猜想。使用分析数理论的圆圈方法动机的完全不同的方法，我们证明了满足Hall-Paige条件的任何组$ g $ $ g $的完整映射的数量是$（e^{ - 1/2} + o（1） + o（1））\，| \，n！^2/n^n $。
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 $(e^{-1/2} + o(1)) \, |G^\text{ab}| \, n!^2/n^n$.
