Authors: Dr Vinit Kumar Sharma, Insaf Ali
Abstract: Group Theory is one of the fundamental branches of modern algebra and provides the mathematical framework for studying symmetry, transformations, and algebraic structures. This paper presents a comprehensive study of groups and their fundamental properties, including closure, associativity, identity, inverses, commutativity, cyclicity, normality, solvability, and nilpotency. The paper examines finite and infinite groups, discusses important classification theorems, and highlights applications in cryptography, coding theory, physics, and computer science. A graphical analysis is presented to compare major classes of groups according to their structural complexity. Recent developments in finite group classification and graph-theoretic approaches to group theory are also reviewed.
DOI: http://doi.org/10.5281/zenodo.20759238
