Introduction of Several Special Groups and Their Applications to Rubik’s Cube

Abstract

Group theory is the subject that aims to study the symmetries and structures of groups in mathematics. This work provides an introduction to group theory and explores some potential applications of group theory on complex geometric objects like the Rubik's cube. To this end, the concepts of symmetric group, permutation group, and cyclic group are introduced, and the famous Lagrange’s theorem and Cayley’s theorem are mentioned briefly. The former theorem establishes that a subgroup’s order must be a divisor of the parent group’s order. Concerning the permutation group, it is a set of permutations that form a group under composition. Hence, the various groups that can be formed by the Rubik's cube are discussed, including the group of all possible permutations of the cube's stickers, and the subgroups that are generated through permutations of the six basic movements embedded in Rubik’s cube. Overall, this essay provides an accessible introduction to group theory and its applications to the popular Rubik's cube.