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.