How is group theory used to represent moves on a Rubik’s Cube?

In modern algebra, group theory provides a rigorous mathematical framework for analyzing the Rubik’s Cube. The cube’s moves are represented as elements of a group, where each rotation of a layer acts as a transformation on the cube's configuration.12

These moves satisfy the fundamental axioms of a group:
* Closure: Performing one rotation followed by another results in a new configuration that is also a valid state of the cube.1
* Associativity: The order in which sequences of moves are grouped does not change the final outcome.1
* Identity: The "identity element" corresponds to the state where no moves are made, leaving the cube unchanged.1
* Inverses: Every rotation has an inverse move (a rotation in the opposite direction) that restores the cube to its previous state.1

By treating the 43 quintillion possible configurations as elements within this group, mathematicians use group theory to develop systematic algorithms for solving the puzzle, effectively navigating the complex state space of the cube.32