# Paige, Lowell J.

## Hall-Paige conjecture ★★★

Author(s): Hall; Paige

A \emph{complete map} for a (multiplicative) group $G$ is a bijection $\phi : G \rightarrow G$ so that the map $x \rightarrow x \phi (x)$ is also a bijection.

\begin{conjecture} If $G$ is a finite group and the Sylow 2-subgroups of $G$ are either trivial or non-cyclic, then $G$ has a complete map. \end{conjecture}

Keywords: complete map; finite group; latin square