Rubik's Cube group

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A Rubik's Cube

In mathematics, the Rubik's Cube is an interesting object because it provides a tangible representation of a mathematical group. The Rubik's Cube group can be thought of as the set of all cube operations with composition as the group operation. Any set of operations which results in a solved cube should be thought of as the identity transformation (the operation that does nothing).

[edit] Formal description

Formally, the Rubik's Cube group can be defined as a permutation group. A 3×3×3 Rubik's cube consists of 6 faces, each with 9 colored squares called facets for a total of 54 facets. However, the 6 facets in the center of the faces are not moved by any cube operation and may be regarded as fixed in space. The cube operations consist of rotating the 6 faces and thereby permuting the remaining 48 facets. The cube group G can then be defined as the subgroup of the full symmetric group S₄₈ generated by the 6 face rotations.

By definition, each element of the cube group is a permutation of the 48 movable facets. However, there is a one-to-one correspondence between elements of the cube group and positions of the Rubik's cube. Any element of the cube group is a permutation that when applied to the solved cube results in a (legal) cube position. Conversely, any legal cube position must be the result of some sequence of face rotations applied to the solved cube, and any such sequence is an element of the cube group.

The order of the cube group G is then equal to the number of possible positions attainable by the cube. This is:

$|G| = \frac{1}{12}8!\;3^8\,12!\;2^{12} = 43,252,003,274,489,856,000$

which factorizes as

$|G| = 2^{27}\,3^{14}\,5^3\,7^2\,11^1.$

Because of the large size of the cube group it is sometimes useful to analyse the structure with the assistance of a computer algebra system such as GAP.

[edit] Structure

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Let Cube be the group of all legal cube operations. In the following, we assume the notation described in How to solve the Rubik's Cube. Also we assume the orientation of the six centre pieces to be fixed.

We consider two subgroups of Cube: First the group of cube orientations, C_o, which leaves every block fixed, but can change its orientation. This group is a normal subgroup of the Cube group. It can be represented as the normal closure of some operations that flip a few edges or twist a few corners. For example, the normal closure of the following two operation is C_o:

BR'D²RB'U²BR'D²RB'U², (twist two corners)

RUDB²U²B'UBUB²D'R'U', (flip two edges)

For the second group we take Cube permutations, C_p, which can move the blocks around, but leaves the orientation fixed. For this subgroup there are more choices, depending on the precise way you fix the orientation. One choice is the following group, given by generators: (The last generator is a 3 cycle on the edges).

C_p = [U², D², F, B, L², R², R²U'FB'R²F'BU'R² ]

Since C_o is a normal subgroup, the intersection of Cube orientation and Cube permutation is the identity, and their product is the whole cube group, it follows that the cube group is the semi-direct product of these two groups. That is

Cube = C_o

C_p.

(For technical reasons, the above analysis is not correct. However, the possible permutations of the cubes, even when ignoring the orientations of the said cubes, is no bigger than C_p, and this means that the cube group is the semi-direct product given above.)

Next we can take a closer look at these two groups. C_o is an abelian group, it is $\mathbb Z_3^7 \times \mathbb Z_2^{11}$ .

Cube permutations, C_p, is little more complicated. It has the following two normal subgroups, the group of even permutations on the corners $A 8$ and the group of even permutations on the edges $A 12$ . Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that

$C_p = (A_8 \times A_{12})\,$

$\mathbb Z_2$

Putting all the pieces together we get that the cube group is isomorphic to

$(\mathbb Z_3^7 \times \mathbb Z_2^{11})$

$\,((A_8 \times A_{12})$

$\mathbb Z_2)$ .

This group can also be described as the subdirect product [(Z₃⁷S₈)×(Z₂¹¹ S₁₂)]½, in the notation of Griess. It does not make sense to take the possible permutations of the centre pieces into account, as this is simply an artefact of the orientation of the cube in Euclidean 3D-space. Rotations of the centre pieces are unimportant on the standard cube, but are crucial when considering non-standard incarnations of the "cube" such as Rubik's calendar and Rubik's world. When all these centre piece symmetries are taken into account, the symmetry group is a subgroup of [Z₄⁶×(Z₃⁷S₈)×(Z₂¹¹ S₁₂)]½ — the editor of this paragraph currently has no inclination to determine whether this is the full symmetry group in this situation. (This unimportance of centre piece rotations is an implicit example of a quotient group at work, 'shielding' the reader from the full automorphism group of the object in question.)

The symmetry group of the Rubik's cube obtained by dismembering it and reassembling is slightly larger: namely it is the direct product $\mathbb Z_4^6 \times \mathbb Z_3\wr \mathrm{S}_8 \times \mathbb Z_2\wr \mathrm{S}_{12}$ . The first factor is accounted for solely by rotations of the centre pieces, the second solely by symmetries of the corners, and the third solely by symmetries of the edges. The latter two factors are examples of wreath products.

The simple groups that occur as quotients in the composition series of the standard cube group (i.e. ignoring centre piece rotations) are $A_8,A_{12},\mathbb{Z}_3\ (7\ \mbox{times}),\mathbb{Z}_2\ (12\ \mbox{times})$ .

[edit] References and external links

David Joyner (2002). Adventures in Group Theory : Rubik's Cube, Merlin's Machine, and Other Mathematical Toys. The Johns Hopkins University Press. ISBN 0-8018-6947-1.. Notes from an earlier version of this book are available online at [1].
Analyzing Rubik's Cube with GAP

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Categories: Cleanup from May 2006 | Rubik's Cube | Finite groups | Permutation groups