A STUDY ON THE STRUCTURE OF ENDOMORPHISMS AND AUTOMORPHISMS IN FINITE ABELIAN GROUPS

Abstract

This paper investigate the algebraic structure of endomorphism and automorphisms within finite Abelian groups. We analyze how the breakdown into direct sums of cycles groups influences the behavior of endomorphisms and their classification. We define End(G) as a subring of the product of matrix rings over modular integers. An explicit formula for |????????????(????????)| is derived ,enabling computation of automorphism counts for any finite Abelian group. we focus on developing a modern and accessible characterization of their automorphism groups. We identify the automorphisms as those matrices that remain invertible modulo a prime. A key result shows that the automorphism group Aut(G) decomposes naturally over relatively prime components. Furthermore, To determine how many automorphisms for every finite abelian groups,connecting linear algebraic properties over finite fields to group-theoretic structure. This study provides both theoretical insights and computational tools relevant to algebraic structures and their symmetries.