Rubik's cube group

Group theory
Group theory
Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (Semi-)direct product
Finite groups Classification of finite simple groups Cyclic group Zn Symmetric group Sn Dihedral group Dn Alternating group An Mathieu groups M11, M12, M22, M23, M24 Conway groups Co1, Co2, Co3 Janko groups J1, J2, J3, J4 Fischer groups F22, F23, F24 Baby Monster group B Monster group M
Discrete groups and lattices Integers Z Lattice Modular groups PSL(2,Z), SL(2,Z)
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 · F4 · E6 · E7 · E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞), SU(∞), Sp(∞)

The Rubik's Cube group is a finite group which corresponds to the set of all cube moves on the Rubik's Cube with composition of cube moves as the group operation. The solved position is just one legal position; there is a one-to-one correspondence between each of the legal positions of the Rubik's cube and the elements of the Rubik's cube group set.

Cube positions and moves

An original 3×3×3 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 basic cube move rotates one of the 6 faces. The face's center facet rotates about its axis but otherwise it stays in the same position; the face's remaining 8 outside facets along with their 12 neighboring facets on the each of the 4 attached faces all change positions; a total of 20 facets change position after a basic cube move while 34 facets don't change position. The 48 facets touching an edge can change position; the other 6 center facets never change their position.

Cube moves are described using the Singmaster notation:

$F$ 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 composition of $LLLL$ is the same as $E$. $RRR$ is the same as $R^\prime$.

Group properties

The Rubik's Cube group G consists of the set of cube moves and the operation of composition. The group properties of G are:

1. associativity, composition is always associative.
2. closure, the composition of two moves is another move.
3. identity element, the empty move $E$ composed in either order with any other move $X$ is the same as $X$.
4. inverse element, reversing a move returns the cube to its previous position. Reversing a sequence of moves inverts the sequence.

Any element of G is a sequence of cube moves which 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 composition is an element of G.

G is a permutation group. The basic cube moves form a generating set, which in turn generates a permutation group by composition.

G is non-Abelian. For example, since $BU$ is not the same as $UB$, not all cube moves are commutative. Some moves are commutative, for example $FB$ is the same as $BF$ and in fact some subgroups of G are Abelian.

The order of G is finite but large. Still, each position can be solved in at most 20 moves:

$|G| = 43{,}252{,}003{,}274{,}489{,}856{,}000\,\!$

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.

G can be defined as the subgroup of the full symmetric group S₄₈ generated by the 6 face rotations.

We consider two subgroups of G: 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 G. It can be represented as the normal closure of some move 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).

For the second group we take G permutations, C_p, which can move the blocks around, but leave 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):

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

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

$G = C_o \rtimes C_p.$

(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 at the same time no bigger than C_p and at least as big as 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 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

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

References and external links

• Structure mathématique du Rubik's Cube.