Rubik's Cube group

Summary

The Rubik's Cube group $(G, \cdot )$ represents the mathematical structure of the Rubik's Cube mechanical puzzle. Each element of the set $G$ corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position. Indeed, with the solved position as a starting point, there is a one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of $G$. The group operation $\cdot$ is the composition of cube moves, corresponding to the result of performing one cube move after another.