Reduction of the Hall–Paige conjecture to sporadic simple groups

A complete mapping of a group G is a permutation such that g↦gϕ(g) is also a permutation. Complete mappings of G are equivalent to transversals of the Cayley table of G, considered as a Latin square. In 1953, Hall and Paige proved that a finite group admits a complete mapping only if its Sylow-2 subgroup is trivial or noncyclic. They conjectured that this condition is also sufficient. We prove that it is sufficient to check the conjecture for the 26 sporadic simple groups and the Tits group.