Determinant for automorphisms of semidirect product of groups

TL;DR

This paper explores the concept of determinants for endomorphisms of semidirect product groups, providing a characterization of invertible endomorphisms using matrix representations and determinant tools.

Contribution

It introduces a determinant concept for endomorphisms of semidirect product groups and characterizes invertible endomorphisms through this framework.

Findings

Determinant for endomorphisms of semidirect product groups is defined.

Invertibility of endomorphisms characterized via determinants.

Matrix representation aids in understanding group endomorphisms.

Abstract

A description of the endomorphisms of semidirect products of two groups as a group of $2\times 2$ matrices of maps is already known. Using this description, we have studied the concept of determinant for the endomorphisms of semidirect product of two groups. A characterization of the invertible endomorphisms is given with the help of the tools developed using the determinants.