Hall and Paige conjectured in 1955 that a finite group G has a complete mapping (a permutation F such that xF(x) is also a permutation) if and only if it satisfies a straightforward necessary condition. This was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. I will discuss joint work with Sean Eberhard and Freddie Manners in which we approach the problem in a more analytic way that enables us to asymptotically count complete mappings.
This is a seminar of the Combinatorial Mathematics Society of Australasia.
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