Hall-Paige conjecture

Statement
Suppose $$G$$ is a finite group with the property that every $$2$$-Sylow subgroup of $$G$$ is either trivial or non-cyclic. Then, there exists a complete map from $$G$$ to $$G$$: a bijection $$\varphi:G \to G$$ such that the map $$g \mapsto g\varphi(g)$$ is also a bijection.

Related conjectures

 * Snevily's conjecture