Determinant for automorphisms of Zappa-Szép product of groups

Abstract

A description of the endomorphisms of Zappa-Szép products of two groups as a group of $2\times 2$ matrices of maps is discussed in \cite{detsem}. Using this description, we have studied the concept of determinant for the endomorphisms of Zappa-Szép products of two groups. A characterization of the invertible endomorphisms is given with the help of the tools developed using the determinants. The determinants are also computed for the alternating group $A_5$.