Rubik's Cube group

The manipulations of the Rubik's Cube form the Rubik's Cube group.

The Rubik’s Cube group is a group $(G, \cdot )$ that corresponds to the set $G$ of all cube moves on the Rubik's Cube mechanical puzzle with the group operation $\cdot$ being the concatenation of cube moves. 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 $G$.^[1][2]

Cube moves

A $3 \times 3 \times 3$ Rubik's Cube consists of $6$ faces, each with $9$ colored squares called facets, for a total of $54$ facets. A solved cube has all of the facets on each face having the same color.

A cube move rotates one of the $6$ faces: $90^\circ , 180^\circ$ or $-90^\circ$ (half move metric).^[3] A center facet rotates about its axis but otherwise stays in the same position.^[1]

Cube moves are described with the Singmaster notation:^[4]

Basic 90°	180°	-90°
$F$ turns the front clockwise	$F^2$ turns the front clockwise twice	$F^\prime$ turns the front counter-clockwise
$B$ turns the back clockwise	$B^2$ turns the back clockwise twice	$B^\prime$ turns the back counter-clockwise
$U$ turns the top clockwise	$U^2$ turns the top clockwise twice	$U^\prime$ turns the top counter-clockwise
$D$ turns the bottom clockwise	$D^2$ turns the bottom clockwise twice	$D^\prime$ turns the bottom counter-clockwise
$L$ turns the left face clockwise	$L^2$ turns the left face clockwise twice	$L^\prime$ turns the left face counter-clockwise
$R$ turns the right face clockwise	$R^2$ turns the right face clockwise twice	$R^\prime$ turns the right face counter-clockwise

The empty move is $E$. The concatenation of $LLLL$ is the same as $E$, and $RRR$ is the same as $R^\prime$.

Group structure

The following uses the notation described in How to solve the Rubik's Cube. The orientation of the six centre facets is fixed.

We can identify each of the six face rotations as elements in the symmetric group on the set of non-center facets. More concretely, we can label the non-center facets by the numbers 1 through 48, and then 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 $|G| = 43{,}252{,}003{,}274{,}489{,}856{,}000\,\! = 2^{27} 3^{14} 5^3 7^2 11$.^[5] 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^[3] (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^[6]).

The largest order of an element in G is 1260. For example, one such element of order 1260 is $(RU^2D^{-1}BD^{-1})$.^[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.^[2]

Subgroups

We consider two subgroups of G: First the subgroup C_o 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:

$B R^\prime D^2 R B^\prime U^2 B R^\prime D^2 R B^\prime U^2,\,\!$ (twist two corners)

$R U D B^2 U^2 B^\prime U B U B^2 D^\prime R^\prime U^\prime,\,\!$ (flip two edges).

Second, we take the subgroup $C_P$ 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.^{[note 1]} One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):

$C_p = [U^2, D^2, F, B, L^2, R^2, R^2 U^\prime F B^\prime R^2 F^\prime B U^\prime R^2].\,\!$

Since C_o is a normal subgroup, the intersection of C_o and C_p 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

$G = C_o \rtimes C_p. \,$

Next we can take a closer look at these two groups. The structure of C_o is

$\mathbb Z_3^7 \times \mathbb Z_2^{11},\$

since the group of rotations of each corner (resp. edge) cube is $\mathbb Z_3$ (resp. $\mathbb Z_2$), and in each case all but one may be rotated freely, but these rotations determine the orientation of the last one. Noticing that there are 8 corners and 12 edges, and that all the rotation groups are abelian, gives the above structure.

Cube permutations, C_p, is a little more complicated. It has the following two normal subgroups, the group of even permutations on the corners A₈ and the group of even permutations on the edges A₁₂. Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that

$C_p = (A_8 \times A_{12})\, \rtimes \mathbb Z_2.$

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

$(\mathbb Z_3^7 \times \mathbb Z_2^{11}) \rtimes \,((A_8 \times A_{12}) \rtimes \mathbb Z_2).$

This group can also be described as the subdirect product $[(\mathbb Z_3^7 \rtimes \mathrm S_8) \times (\mathbb Z_2^{11} \rtimes \mathrm{S}_{12})]^\frac{1}{2}$, in the notation of Griess.

Generalizations

When the centre facet symmetries are taken into account, the symmetry group is a subgroup of

$[\mathbb Z_4^6 \times (\mathbb Z_3^7 \rtimes \mathrm S_8) \times (\mathbb Z_2^{11} \rtimes \mathrm S_{12})]^\frac{1}{2}.$

(This unimportance of centre facet 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 times), and $\mathbb Z_2$ (12 times).

See also

• Commutator
• Conjugacy class -Conjugates
• Coset
• Optimal solutions for Rubik's Cube
• Solvable group
• Thistlethwaite's algorithm

Notes

1. One way of defining orientation is as follows, adapted from pages 314–315 of Metamagical Themas by Douglas Hofstadter. Define two notions: the chief color of a block and the chief facet of a position, where a position means the location of a block. The chief facet of a position will be the one on the front or back face of the cube, if that position has such a facet; otherwise it will be the one on the left or right face. There are nine chief facets on F, nine on B, two on L, and two on R. The chief color of a block is defined as the color that should be on the block's chief facet when the block "comes home" to its proper position in a solved cube. A cube move $X$ preserves orientation if, when $X$ has been applied to a solved cube, the chief color of every block is on the chief facet of its position.

References

1. Joyner, David (2002). Adventures in group theory: Rubik's Cube, Merlin's machine, and Other Mathematical Toys. Johns Hopkins University Press. ISBN 0-8018-6947-1.
2. Davis, Tom (2006). "Group Theory via Rubik’s Cube" (PDF).
3. Rokicki, Tomas; et al. "God's Number is 20".
4. Singmaster, David (1981). Notes on Rubik's Magic Cube. Penguin Books. ISBN 0907395007.
5. Schönert, Martin. "Analyzing Rubik's Cube with GAP".
6. God's Number is 26 in the Quarter-Turn Metric