讲座内容：

A complete mapping of a finite group $G$ is a permutation $\phi: G\rightarrow G$ such that $x\mapsto x\phi(x)$ is also a permutation. A permutation $\theta:G\rightarrow G$ of a finite group $G$ is an orthomorphism of $G$ if the mapping $x\mapsto x^{-1}\theta(x)$ is also a permutation. Two orthomorphisms $\theta$ and $\phi$ of $G$ are orthogonal if the mapping $x\mapsto \theta(x)^{-1}\phi(x)$ is bijective. This talk provides a brief introduction to the existence of complete mappings and orthogonal orthomorphisms of groups, focusing on their algebraic and extremal aspects. Difference matrices have a close relationship with orthogonal orthomorphisms of groups. It will also give a survey on difference matrices and their related topics, such as orthogonal arrays and mutually orthogonal Latin squares.