The branch of mathematics that is devoted to the study of groups is called categories theory. It is one of the most important algebraic structures that play a basic role in abstract algebra, and in various sciences, physics, quantum, and - is of paramount importance. The idea of forming group¿s theory was formed when mathematicians observed that the structures they study are in common properties, and if they can examine all these properties on a certain structure, they actually study a wide range of similar structures and thus save time. The theory of groups was developed by major composers of classical algebra, numbers theory, geometry, and analysis. The theory of numbers was studied and expanded in 2006. Although the concept of transformations is slowly developed in the study of geometry, the main task in extending the concept of the group is derived from the study of polynomial equations.