Basic Concept of Isomorphism and its Use in Group Theory and Physics

Abstract

Isomorphism is a central concept in mathematics and other sciences for defining structures and their equivalencies. The aim of this paper would be to describe the historical commencement of group theory and introduce the basics of concepts in isomorphisms. This paper defines some of the fundamental theorems related to isomorphisms, such as three isomorphism theorems and the epimorphism lemma, whereby the notion of how mathematical objects could have maps onto each other in a structure-preserving way can be defined. It also summarizes the practical applications of isomorphisms in mathematics and physics. Isomorphisms play a huge role in K-theory-a branch of mathematics that deals, in essence, with vector bundles and modules-and in quantum mechanics to further help in understanding symmetries and transformations in physical systems. The importance and implication of isomorphism are very far-reaching in unifying the concepts across scientific fields. This work underscores the importance of isomorphism and related concepts in group theory and other subjects.