Complete mappings

Today I gave a short talk about complete mappings. A complete mapping of a group is a bijection on the group such that the positions in the Cayley table corresponding to an element and its image select a transversal. In other words, the cells so specified include one in each row, one in each column, and one containing each letter. The idea goes back not far short of 100 years. Hall and Paige conjectured that a finite group has a complete mapping if and only if its Sylow 2-subgroups are trivial or non-cyclic. They took a few steps towards a proof, but the conjecture was not settled until 2009, with the work of Stuart Wilcox (who reduced it to looking at simple groups and dealt with the groups of Lie type), Tony Evans (who did most of the sporadic simple groups) and John Bray (who did the final case, the fourth Janko group). In this decade there were two remarkable new approaches to the problem. In 2021, Sean Eberhard, Freddie Manners and Rudi Mrazović posted on the arXiv a paper in…