Complete map

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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