RUBIKA ESOTERICA
This chapter is a brief summary of various facts about the
Rubik's cube group. The material has been taken from [B]
and [BH].
Let G < S54 be the group of moves of the Rubik's cube.
1. |G| = 2^(27)*3^(14)*5^3*7^2*11 = (4.3...)*10^(19).
2. G is generated (as a permutation group) by
m991 = U*B*L*U*L^(-1)*U^(-1)*B^(-1)
and
m992 = R^2*F*L*D^(-1)*R^(-1).
3. The slice subgroup. A "middle slice" is one of the three sets of 8
subcubes each all lying on a plane parallel to a face. The "basic
slice moves" are
MR = middle right slice rotation by 90 degrees
(viewed from the right face),
MF = middle front slice rotation by 90 degrees
(viewed from the front face),
MU = middle up slice rotation by 90 degrees
(viewed from the up face).
Let
S = < MR, MU, MF >
denote the "slice group" generated by the basic slice moves.
Note that this group leaves the corner subcubes fixed but not the
center facets.
Theorem: |S| = 768.
Theorem: S = { g in G | g_V = 1, sgn(g_E) = 1, x_g = (0,...,0) }.
4. The center of G is given by
Z(G) = {1, m490}
where m490 is the "superflip" which leaves all corners alone
and flips every edge:
m490 =
R*L*F*B*U*D*R*L*F*B*U*F^2*MR*F^2*U^(-1)*MR^2*B^2*MR^(-1)*B^2*U*MR^2*D
5. (a) Every group H of order less than 13 is isomorphic to a
subgroup of G.
(b) Every non-abelian group H of order less than 26 is isomorphic
to a subgroup of G.
(c) Z/13Z (the cyclic group of order 13) is not isomorphic to
a subgroup of G.
(d) D26 (the dihedral group of order 26) is not isomorphic to
a subgroup of G.
6. Let Q denote the quaternion group:
Q = {1, -1, i, -i, j, -j, k, -k},
where i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j, and in general,
xy = -yx for x,y belonging to i,j,k. Then Q is isomorphic to the group
Q* = <1, m435, m706, m707, m710> < G,
where
m435 = F^2*MR^(-1)*F^2*MR^2*U^(-1)*MR^2*F^2*MR^2*F^2,
m706 = F^2*MR*U^(-1)*MR^(-1)*U^(-1)*MR*U*MR^(-1)*U*F^2,
m707 = B^(-1)*F^2*R^(-1)*U^(-1)*MR*U*R*U*MR^(-1)*U^(-1)*F^2*B,
m710 = F*U^2*F^(-1)*U^(-1)*L^(-1)*B^(-1)*U^2*B*U*L.
7. This is a long one.
Definition: The "commutator subgroup" G' of G is the subgroup consisting
of all finite products of commutators
[g,h]=g*h*g^(-1)*h^(-1),
where g,h are arbitary elements of G.
Theorem: |G'|=|G|/2.
In fact, we can determine G' (and, while we're at it, G) "exactly".
First, some preliminaries. We identify each g in G with a 4-tuple
(rho(g),sigma(g),x,y),
where
rho(g) = corresponding permutation of the set of vertices of the
cube,
sigma(g) = corresponding permutation of the set of edges of the cube,
and x,y are "orientations" defined below.
To define these orientations, we must first make some choices. Assume
for the moment that the cube is fixed in space in the "solved" position.
For each moveable subcube, choose once and for all a facet on that
subcube. There are three possible choices for each corner subcube, two
for the edges and none for the centers. Mark each of these facets with an
imaginary '+'. (Incidently, choosing a side of each center facet leads to
what is sometimes called the 'supercube', which we will not discuss here.
The idea is that to solve the 'supercube' you must get all the facets back
in the solved position and get the center facets back in their choosen
orientation. This is equivalemnt to superimposing a snapshot of someone
on each face of the cube with their nose on the center facet. To solve this
puzzle you must have all the noses lined up - see [B]).
Now we can define x,y. Label the 8 vertices - and hence also the 8 corner
subcubes - with the numbers 1, 2, ..., 8. Likewise label the 12 edges -
and hence also the 12 edge subcubes - with the numbers 1, 2, ..., 12. Each
move of the Rubik's cube corresponds to an element g of the Rubik's cube
group G. Each g in G yields
a) a permutation rho(g) of the 8 corner subcubes,
b) a permuation sigma(g) of the 12 edge subcubes,
c) for each edge subcube, either
(i) a '0' if the '+ facet' for that subcube when it was in the solved
position is sent to the '+ facet' for that subcube when it was in
the present position,
(ii) a '1' otherwise,
thus yielding a 12-tuple of 0's and 1's: y_g = (y1,y2,...,y12),
d) for each corner subcube, either
(i) a '0' if the '+ facet' for that subcube when it was in the solved
position is sent to the '+ facet' for that subcube in the present
position,
(ii) a '1' if the '+ facet' for that subcube when it was in the solved
position is sent to the facet which is a 120 degrees rotation about
its vertex from the '+ facet' for that subcube in the present
position,
(ii) a '2' otherwise,
thus yielding a 8-tuple of 0's, 1's, and 2's: x_g = (x1,x2,...,x8).
Example: Suppose that we label all the edge and corner facets on the front
with a '+'. The "monoswap"
ms = F*D*F^2*D^2*F^2*D^(-1)*F^(-1)
permutes the top uf corners, twisting the FRU corner 120 degrees clockwise
and the FLU corner counterclockwise. The FRU vertex gets a '1' associated
to it and the FRU vertex gets a '2', for example.
(It also messes up some parts of the rest of the cube.)
Question: Given a 4-tuple (r, s, x, y), where r, s are permutations
as above and
x in {0, 1, 2}^8, y in {0, 1}^(12),
what conditions on r, s, x, y insure that it corresponds to a possible
position of the Rubik's cube?
Theorem: A 4-tuple (r, s, x, y) as above (r in S8, s in S12, etc)
corresponds to a possible position of the Rubik's cube if and only if
(a) sgn(r) = sgn(s), ("equal parity as permutations")
(b) x1 + ... + x8 equals 0 (mod 3), ("conservation of total twists")
(c) y1 + ... + y12 equals 0 (mod 2), ("conservation of total flips").
Corollary: G = { g in G | (a), (b), (c) in the above theorem hold }.
Notation: If g in G then we write the corresponding position as
(rho(g), sigma(g), x_g, y_g).
Theorem: Let Sn denote the symmetric group on n letters.
(a) rho : G --> S8 is a homomorphism,
(b) sigma : G --> S12 is a homomorphism.
Finally, we can describe the commutator subgroup:
Theorem: G' = { g in G | sgn(rho(g)) = sgn(sigma(g)) = 1 }.
This implies that |G/G'| = 2.
8. If g and h belong to G then
x_(gh) = x_g + rho(g)(x_h),
y_(gh) = y_g + sigma(g)(y_h).
9. Let
H = {(r, s, x, y) | r in S8, s in S12,
x = (x1, x2, ..., x8), xi in {0, 1, 2},
x1 + ... + x8 equals 0 (mod 3),
y = (y1, y2, ..., y12), yi in {0, 1},
y1 + ... + y12 equals 0 (mod 2) }.
Define a binary operation * : HxH --> H by
(r,s,x,y)*(r',s',x',y') = (r*r',s*s',x + r(x'), y + s(y')).
This defines a group structure on H.
Theorem: There is an isomorphism between H and
(Z2 wr S12)x(Z3 wr S8),
where Zn is the cyclic group with n elements and wr denotes the wreath
product. In particular,
|H| = |S8||S12||Z2^(12)||Z3^8| = 8!12!2^(11)3^7.
Theorem: The Rubik's cube group G is the kernel of the homomorphism
phi : H ---> {1, -1}
(r,s,x,y) |--> sgn(r)sgn(s).
In particular, G < H is normal of index 2 and
|G| = 8!12!2^(10)3^7.
10. Let
N = { g in G | sgn(sigma(g)) = 1, sgn(rho(g)) = 1 }.
By a theorem above, N = G'.
Theorem: N is a normal subgroup of G,
(b) G/N is a group of order 2.