A short homage to group theory
Group theory is the main building block when it comes to the understanding the concepts of abstract algebra. This specifically concerns the further abstraction to other algebraic structures such as ring, modules and fields when provided with further axioms and operations. This at first sight might seem like a very abstract topic, requiring a profound mathematical background. However this becomes more easily when relating the symmetric properties to a rubic cube.
Rubic cubes were first invented 1974 by Ernö Rubic made public available in 1980. Since that 350 million cubes were sold, further leading to development of algorithms for solving it [1,2]. While competitions have evolved in solving the rubic cube within the shortest period of time, new interest has lead to soliving the cube within the minimum number of moves. This number has been determined to be 20, also often also called the God’s Number [3].
From a mathematical point of view, the different configurations comprising of both vertical and horizontal rotations make up a form a subgroup of a permutation group. For this it is assumed for the rubic cube group (S,◦) that the set of elements correspond to a move (or rotation). However, to get a better understanding lets first get an understanding what a group is.
A group represents a set consisting of a binary operation ◦ combining two elements to form another element. This assumes that the group axioms are satisfied: closure, associativity, identity and invertability. For this let S be a set satisfying the following criteria:
- For all x,y ϵ S, x ◦ y ϵ S(closure).
- Assuming that x, y, z ϵ S, we can define the axioms on associativity where (xy)z = x(yz).
- Assuming e ϵ S, there exists an identity element e ◦ x = x ◦ e = x (identity).
- For all xϵ S, where e represents the identity element (inverse) we can state that
In the case that only closure and associativity (1.+2.) is satisfied we are talking about a semigroup. If the axioms closure, associativity and an identity (1.+2.+3.) is satisfied, we call these monoid. When additionally the commutativity of a group is satisfied, such that xy = yx for some x,yϵ S, one speaks of a abelian or commutative group.
Let’s first see if the rubic cube satisfies the group axioms such that (S, ◦) persist. For this it it can be taken for granted that all possible elements of S will represent all the possible moved or rotations to make. Furthermore, two moves are the same if the resulting cube configurations are the same. This is for instance the case, when twisting the cube clockwise by 180° as twisting it anticlockwise by 180°. Thus, the different moves made represent the elements of the set S. But why does this compose a group? For this lets go through the axioms step by step
S is closed under the binary operation ◦, if for the moves m1 and m2, m1 ◦ m2 also makes up a move (closed). Further if we let e be the identity move (no changes are made to the configuration of the Rubic cube), then m1◦e=e◦m=m1. This is straightforward since first doing move m1 and than nothing is equal to the doing nothing and then taking move m1. This is equally true to only exectuing move m1. Thus, the identity axiom is satisfied. Equally easy to show is that the inverse axiom is fullfilled. When doing a move m1 the reverse m1^-1 results it equals to doing nothing.
More difficult becomes showing that the assosciativity axiom is satisfied. A move is determined by the position and orientation to make. Let C be an oriented cubie we can write M(C) after applying a move m. If applying to moves M1 and M2 than M1◦M2 results in first applying M1 after M2. Thus, M1 moves c to to the cubicle M1(C) and followed by a move M2 such that M2(M1(c)). To show that binary operation ◦ is assosciative, we have to show that (M1 ◦ M2) ◦ M3 = M1 ◦ (M2 ◦ M3) for which M1,M2,M3 ϵ G. Based on the previous equation we can write that
Furthermore, we can define
The assosciativity axiom is therefore fullfilled [4].
Rubic cube as a permutation group
A group G is called a permutation group whose elements are permutation of a a given set S whose group operations is the composition of permutations of the group. The elements of G are called permutations of S, also denoted by Perm(S). The set of all permutations of n objects forms a group Sn of order n!. It is called the nth symmetric group denoted by Sym(S) [5]. The permutation group thus remains a subgroup of the symmetric group such that
Mathematically it can be shown that the Rubic Cube is a permutation group. For this have a look at Figure 1. The rubic cube consists of 6 different colors each consisting of 9 cubies. Considering that the cube makes up an ordered list which has 54 elements, we can rotate the 6 faces of the cube so we can define 6 basic operations or permutations leading to a new composition. Repeating and combining these permutations we can define new permutations, which rearrange the list in a different way [6].
Conclusion
This post has explored group theory in the sense of defining a (permutation) group based on rubic square. It was shown that the rubic square obeys the group properties. Furthermore, the rubic square does not show to be abelian since the commutivity axiom is not obeyed. Showing that the rubic square makes up a permutation group, we can also call this a subgroup of a symmetric group.
Literature
[1] Brandelow, Christoph. Inside the Rubik’s Cube and Beyond. Birkhäuser (1982).
[2] Daniels, Lindsay. Group Theory and the Rubik’s Cube, Lakehead University, Ontario, 2014.
[3] Rokicki, Tomas; Kociemba, Herbert; Davidson, Morley; Dethridge, John. The diameter of the Rubik’s Cube is twenty. SIAM J. Discrete Math, 27, No 2 (2013) 1082–1105.
[4] Chen, Janet. Group Theory and the Rubik’s Cube.
[5] Lang, Serge. Algebra. Revised Third Edition. Springer. 2000.
[6] https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group/ , viewed on 10.01.2001.
