International Journal of Research in Engineering and Applied Sciences(IJREAS) Vol. 9 Issue 5, May -2019 ISSN (O): 2249-3905, ISSN(P): 2349-6525 | Impact Factor: 7.196 International Journal of Research in Engineering & Applied Sciences Email:- editorijrim@gmail.com, http://www.euroasiapub.org An open access scholarly, Online, print, peer-reviewed, interdisciplinary, monthly, and fully refereed journal. 9 MAJOR CLASSIFICATION OF GROUP THEORY IN MODERN ALGEBRA: A STUDY Manoj Kumar 1 , Dr. Pardeep Goel 2 Department of Mathematics 1,2 OPJS University, Churu, Rajasthan Abstract The study of groups emerged from the get-go in the nineteenth century regarding the solution of equations. Originally a group was a set of permutations with the property that the combination of any two permutations again has a place with the set. In this way this definition was summed up to the idea of an abstract group, which was defined to be a set, not really of permutations, together with a technique for joining its elements that is liable to a couple of basic laws. The theory of abstract groups has a significant influence in present day mathematics and science. Groups emerge in a confusing number of clearly detached subjects. Accordingly they show up in crystallography and quantum mechanics, in geometry and topology, in examination and algebra, in material science, science and even in science. One of the most significant instinctive ideas in mathematics and science is symmetry. Groups can depict symmetry; undoubtedly a large number of the groups that emerged in mathematics and science were experienced in the study of symmetry. This discloses somewhat why groups emerge so often. 1. OVERVIEW Abstract Algebra is the study of algebraic systems in an abstract way. “We are as of now acquainted with a number of algebraic systems from your prior examinations. For example, in number systems, for example, the integers the rationalnumbers real numbers or the complex numbers there are algebraic operations such as addition,subtraction, and multiplication.” There are comparative algebraic operations on different items - for example vectors can be included or subtracted, 2 x 2 matrices can be included, subtracted and duplicated. A few times these operations fulfill comparable properties to those of the well-known operations on numbers, yet in some cases they don't. For example, in the event that a; b are numbers then we realize that stomach muscle = ba. However, there are examples of 2 x 2 matrices A;B to such an extent that AB ≠BA: