Almost everyone has, at some point, played with the deceptively simple puzzle that is a Rubik’s cube. Even before studying pure maths, we had heard about the idea of Group Theory through an interest in Rubik’s Cubes. Rubik’s Cubes are a common example in group theory because of the physical illustration of the mathematical idea of a group. A group is defined as a set equipped with an operation with elements composed into other elements, identity and inverse elements, as well as being associative. The elements of the group that defines a Rubik’s Cube are all the possible scrambles. The operations are generated by rotating faces of the cube, meaning algorithms that manipulate the cube in more specific ways can always be broken down into a collection of face turns. The identity of the Rubik’s is not moving the cube, which maintains the current scramble. The inverses are just rotating faces in opposite directions or composing those inverses as to undo a collection of moves; this can be thought of as undoing the moves of a scramble to solve the cube again. Finally, associativity can be thought of as the fact that any string of moves can be grouped together into collections of easy to do moves without affecting the algorithm, which is rather important for solving cubes quickly.
The reader may be wondering the exact mathematical construction of a Rubik’s Cube. For example, a clock is used to represent a group called the “cyclic group” because of the repeated cycles through the numbers. Similar to the jump in complexity of movement of the hand of a clock to the movement of pieces of a Rubik’s Cube, the group representing a Rubik’s Cube is too complex for the purpose of this article. For solving a Rubik’s Cube, it is not necessary to know the mechanical workings, but it can help gain a deeper understanding of the cube. The same concept will be applied to the group as the specific construction will not be necessary for following conclusions.
From a strictly mechanical standpoint, a 2x2 Rubik’s Cube is the same as a 3x3. A 2x2 functions as a 3x3 with smaller edges hidden behind larger corners. The same idea applies to the groups that define each of the different cubes. A quotient group is a group that can be obtained through sending elements of a group to the quotient group through an equivalence relation. For the quotient group Q = G/H, Q is constructed by sending elements of G that differ only by elements of H to the same element in Q. If G were to be the group of a 3x3 and we wanted to make Q a 2x2, how do we construct H. Thinking back to the physical way this is done, we want an operation that “hides” the edge pieces and only gives us corner pieces. The necessary group turns out to be the group of algorithms that move and rotate edge pieces of the Rubik’s Cube. The reader should check for understanding of the following: if all of the possible edge cases are equivalent, the equivalence classes can be represented as different possible corner positions, which is just a 2x2.
The last step to confirm that the results are mathematically sound is to confirm that the group of edge positions and orientations is a normal subgroup of the 3x3. A subgroup means that the edge positions are a subset of the total scrambles and that the moves on a 3x3 are moves on the set of edge positions, which can be checked through an understanding of the Rubik’s Cube itself. To be a “normal” subgroup, the edges must follow the property that the combination of any moves on the 3x3, then moving only edges, and then the inverse of the first moves results in moving edges. Since edges and corners are independent of each other, the corners are unaffected by the middle move and thus scrambled and solved by the first and last move. Only edges can change during this process, meaning that the subgroup of edges is normal.
It has taken groups, quotient groups, equivalence relations, and normal subgroups all to mathematically represent that simply ignoring edges on a 3x3 Rubik’s Cube leaves a 2x2. Armed with this knowledge, the reader may consider what other quotient groups can be created on a 3x3 by making other collections of scrambles equivalent. For example, can the group of moves that only move and rotate corners be treated the same why we treated the group of moves that only move and rotate edges?
Contributors: Joseph Brewster, David Staudinger