Group Theory and Solutions of the Rubik’s Cube

Abstract

The paper explores the application of group theory to solving the Rubik’s Cube. Group theory, a branch of abstract algebra, studies sets equipped with an operation that satisfies specific properties: closure, associativity, identity, and inverses. The Rubik’s Cube is treated as a group because its set of moves forms a structure that aligns with these properties. By representing the cube’s various configurations and moves through the mathematical constructs of groups, the paper analyzes how the elements transform during cube manipulations. Key concepts such as symmetric groups, homomorphisms, alternating groups, and disjoint cycle decomposition are used to break down the cube’s complexity. Additionally, the study demonstrates how group actions, particularly the parity of permutations and orientation sums, govern valid cube configurations. The paper also references methodologies from other studies to apply and refine these mathematical approaches for systematically solving the Rubik’s Cube. Through these methods, the research illustrates how the cube can be navigated using logical algorithms grounded in group theory principles