Complete map

Definition
A complete map from a group $$G$$ to itself is a bijection $$\varphi:G \to G$$ such that the map $$g \mapsto g \varphi(g)$$ is also a bijection.

Note that if $$\alpha$$ is an automorphism of $$G$$, the map $$g \mapsto \alpha(g^{-1})$$ is a complete map if and only if $$\alpha$$ is a fixed-point-free automorphism.

Facts

 * Hall-Paige conjecture