A determinant for automorphisms of groups

TL;DR

This paper introduces a determinant concept for automorphisms of product groups, providing a new way to characterize invertible automorphisms and describing the automorphism group explicitly for certain classes.

Contribution

It develops a determinant framework for automorphisms of product groups and characterizes invertibility, offering explicit descriptions and computational advantages.

Findings

Determinant concept for automorphisms of $H\times K$

Characterization of invertible automorphisms using determinants

Explicit description of Aut($H\times K$) as 2x2 matrices

Abstract

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.