A determinant for automorphisms of groups

Let $H$ and $K$ be groups. In this paper we introduce a concept of determinant for automorphisms of $H\times K$ and some concepts of incompatibility for group pairs as a measure of how much $H$ and $K$ are fare from being isomorphic. With the aid of the tools developed from these definitions, we give a characterisation of invertible automorphisms of $H\times K$ by means of their determinants and an explicit description of Aut($H\times K$) as a group of $2$-by-$2$ matrices, in case $H$ or $K$ belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants.