The existence of strong complete mappings of finite groups: A survey

Abstract

A complete mapping of a group $G$ is a bijection $\theta :G\to G$ for which the mapping $x\mapsto x\theta \left(x\right)$ is a bijection, and a strong complete mapping of a group $G$ is a complete mapping $\theta$ of $G$ for which the mapping $x\mapsto x^{-1}\theta \left(x\right)$ is also a bijection. Complete mappings and strong complete mappings have several combinatorial applications. While the existence problem for complete mappings of finite groups has been settled, very little is known about the existence of strong complete mappings of finite groups. We survey the known existence results for strong complete mappings of finite groups and we suggest directions for further work on this existence problem.