Homomorphism, Isomorphism, and Their Applications in Group Theory

Abstract

The main study of modern algebra is the so-called algebraic systems, i.e., sets with operations. Modern algebra has important applications in other branches of mathematics and natural science. In modern algebra, homomorphism and isomorphism are relatively elementary but extremely important concepts that are interrelated and different. A homomorphism is a mapping that preserves the constitution of a mathematical format and is a generalization of a homomorphism. it is a structurally invariant mapping between two algebraic structures in abstract algebra. The conditions for a homomorphism to be a homomorphism in different algebraic systems are different, and the conditions for a homomorphism to be a homomorphism are described here. Some applications of homomorphisms and homomorphisms to different algebraic systems are discussed, from which the importance of homomorphisms and homomorphisms is illustrated.