# Complete map

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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.