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Rubik's Cube Group

By Moloy De posted Thu April 08, 2021 10:19 PM

  
The Rubik's Cube group is constructed by labeling each of the 48 non-center facets with the integers 1 to 48. Each configuration of the cube can be represented as a permutation of the labels 1 to 48, depending on the position of each facet. Using this representation, the solved cube is the identity permutation which leaves the cube unchanged, while the twelve cube moves that rotate a layer of the cube 90 degrees are represented by their respective permutations. The Rubik's Cube group is the subgroup of the symmetric group S48 generated by the six permutations corresponding to the six clockwise cube moves.

With this construction, any configuration of the cube reachable through a sequence of cube moves is within the group. Its operation refers to the composition of two permutations; within the cube, this refers to combining two sequences of cube moves together, doing one after the other. The Rubik’s Cube group is non-abelian as composition of cube moves is not commutative; doing two sequences of cube moves in a different order can result in a different configuration.

We can identify the six face rotations as elements of the symmetric group S48 according to how each move permutes the various facets. The Rubik's Cube group, G, is then defined to be the subgroup of S48 generated by the 6 face rotations F,B,U,D,L,R. The cardinality of G is given by |G| = 43,252,003,274,489,856,000.

Despite being this large, God's Number for Rubik's Cube is 20; that is, any position can be solved in 20 or fewer moves where a half-twist is counted as a single move; if a half-twist is counted as two quarter-twists, then God's number is 26.

The largest order of an element in G is 1260. For example, one such element of order 1260 is (RU^{2}D^{-1}BD^{-1})}(RU^{2}D^{-1}BD^{-1}). G is non-abelian since, for example, FR is not the same as RF. That is, not all cube moves commute with each other.

We consider two subgroups of G: First the subgroup Co of cube orientations, the moves that leave the position of every block fixed, but can change the orientations of blocks. This group is a normal subgroup of G. It can be represented as the normal closure of some moves that flip a few edges or twist a few corners. For example, it is the normal closure of the following two moves:



Second, we take the subgroup CP of cube permutations, the moves which can change the positions of the blocks, but leave the orientation fixed. For this subgroup there are several choices, depending on the precise way you define orientation. One choice is the following group, given by generators:

Since Co is a normal subgroup and the intersection of Co and Cp is the identity and their product is the whole cube group, it follows that the cube group G is the semi-direct product of these two groups. That is
QUESTION I: What was Rubik's intention while devising the Cube?
QUESTION II: Who proved the God Number to be 20 and how?
REFERENCE: Wikipedia

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