Groups are vital for modern algebra; Its basic structure can be found in many mathematical phenomena. Groups can be found in geometry, which represent phenomena such as symmetry and some types of transformations. Group theory has applications in physics, chemistry and computer science, and even puzzles such as the Rubik's Cube can be represented using group theory. In this extended summary, we give the definition of a group and several theorems in group theory. We also have several important examples of groups, namely the permutation group and the symmetry group, together with their applications.Group theory, in modern algebra, the study of groups, which are systems that comprise of a set of components and a binary operation that can be connected to two components of the set, which together fulfill certain adages.
