Reduction of the Hall-Paige conjecture to sporadic simple groups

TL;DR

This paper reduces the proof of the Hall-Paige conjecture to verifying it for 26 sporadic simple groups and the Tits group, simplifying the overall problem.

Contribution

The authors show that confirming the Hall-Paige conjecture for these specific groups suffices to prove it for all finite groups.

Findings

Reduction of the conjecture to sporadic simple groups and Tits group

Verification for these groups implies the conjecture for all finite groups

Simplifies the proof process for the Hall-Paige conjecture

Abstract

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.