Reduction of the Hall-Paige conjecture to sporadic simple groups

A complete mapping of a group $G$ is a permutation $\phi:G\rightarrow G$ such that $g\mapsto g\phi(g)$ is also a permutation. Complete mappings of $G$ are equivalent to tranversals 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 non-cyclic. 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.