Rubik's Cube group

Group structure

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

We can identify each of the six face rotations as elements in the symmetric group on the set of non-center facelets. More concretely, we can label the non-center facelets 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 facelets. The Rubik's Cube group, G, is then defined to be the subgroup of S₄₈ generated by the 6 face rotations, ${\displaystyle \{F,B,U,D,L,R\}}$.

The cardinality of G is given by:

${\displaystyle |G|=\mathrm{43,252,003,274,489,856,000}}$${\displaystyle =\frac{12!\cdot 8!}{2}\cdot \frac{2^{12}}{2}\cdot \frac{3^8}{3}}$${\displaystyle =2^{27}{3}^{14}{5}^3{7}^{2}11}$

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

${\displaystyle (R{U}^2{D}^{-1}B{D}^{-1})}$.

G is non-abelian (that is, not all cube moves commute with each other) since, for example, ${\displaystyle FR}$ is not the same as ${\displaystyle RF}$. The center of G consists of only two elements: the identity (i.e. the solved state), and the superflip.

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:

${\displaystyle B{R}^{\mathrm{\prime }}{D}^{2}R{B}^{\mathrm{\prime }}{U}^{2}B{R}^{\mathrm{\prime }}{D}^{2}R{B}^{\mathrm{\prime }}{U}^2,}$ (twist two corners)

${\displaystyle RUD{B}^2{U}^2{B}^{\mathrm{\prime }}UBU{B}^2{D}^{\mathrm{\prime }}{R}^{\mathrm{\prime }}{U}^{\mathrm{\prime }},}$ (flip two edges).

Second, we take the subgroup ${\displaystyle C_{\text{p}}}$ 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 'orientation' is defined. One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):

${\displaystyle C_{\text{p}}=[U^2,D^2,F,B,L^2,R^2,R^2{U}^{\mathrm{\prime }}F{B}^{\mathrm{\prime }}{R}^2{F}^{\mathrm{\prime }}B{U}^{\mathrm{\prime }}{R}^2].}$

Since C_o is a normal subgroup and 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

${\displaystyle G=C_{\text{o}}⋊C_{\text{p}}.}$

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

${\displaystyle {\mathrm{Z}}_3^7\times {\mathrm{Z}}_2^{11},}$

since the group of rotations of each corner (resp. edge) cube is ${\displaystyle {\mathrm{Z}}_3}$ (resp. ${\displaystyle {\mathrm{Z}}_2}$), 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 disjoint 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 subgroups is a permutation that swaps two corners and swaps two edges. It turns out that these generate all possible permutations, which means

${\displaystyle C_{\text{p}}=({\mathrm{A}}_8\times {\mathrm{A}}_{12})⋊{\mathrm{Z}}_2.}$

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

${\displaystyle ({\mathrm{Z}}_3^7\times {\mathrm{Z}}_2^{11})⋊(({\mathrm{A}}_8\times {\mathrm{A}}_{12})⋊{\mathrm{Z}}_2).}$

This group can also be described as the subdirect product

${\displaystyle [({\mathrm{Z}}_3^7⋊{\mathrm{S}}_8)\times ({\mathrm{Z}}_2^{11}⋊{\mathrm{S}}_{12}){]}^{\frac{1}{2}},}$

in the notation of Griess.

Generalizations

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

${\displaystyle [{\mathrm{Z}}_4^6\times ({\mathrm{Z}}_3^7⋊{\mathrm{S}}_8)\times ({\mathrm{Z}}_2^{11}⋊{\mathrm{S}}_{12}){]}^{\frac{1}{2}}.}$

(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 disassembling and reassembling it is slightly larger: namely it is the direct product

${\displaystyle {\mathrm{Z}}_4^6\times ({\mathrm{Z}}_3\wr {\mathrm{S}}_8)\times ({\mathrm{Z}}_2\wr {\mathrm{S}}_{12}).}$

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 generalized symmetric groups, which are themselves examples of wreath products. (There is no factor for re-arrangements of the center faces, because on virtually all Rubik's Cube models, re-arranging these faces is impossible with a simple disassembly.)

The simple groups that occur as quotients in the composition series of the standard cube group (i.e. ignoring centre piece rotations) are ${\displaystyle {\mathrm{A}}_8}$, ${\displaystyle {\mathrm{A}}_{12}}$, ${\displaystyle {\mathrm{Z}}_3}$ (7 times), and ${\displaystyle {\mathrm{Z}}_2}$ (12 times).

Conjugacy classes

It has been reported that the Rubik's Cube group has 81,120 conjugacy classes. The number was calculated by counting the number of even and odd conjugacy classes in the edge and corner groups separately and then multiplying them, ensuring that the total parity is always even. Special care must be taken to count so-called parity-sensitive conjugacy classes, whose elements always differ when conjugated with any even element versus any odd element.