Rubik's Cube Group

Basics: 

• A 3x3x3 Rubik's Cube has 6 faces. 
• Each face consists of 9 stickers and every sticker color is the one from the set {red, blue, green, orange, white, yellow}.
• There are total 12 edges in the cube.

Solved cube: 

• A cube is said to be solved if all the stickers in a face are of the same color.

Cube Move: 

• A move which rotates any one face by either 90⁰, -90⁰ or 180⁰.

Let M be any move. Then,

M : turn the face M clockwise

M' : turn the face M anti-clockwise

where, M belongs to {F, B, U, D , L , R}

Center Sticker: 

• A center sticker always either remains stationary or move about its axis at the same position.

Let, G = set of all possible cube moves in a 3x3x3 Rubik's Cube

(Blindfolded Rubik's Cube Tutorial)

Cardinality of G:

• Number of ways in which 8 corners can be permuted (without changing orientation) = 8!

 

• Number of ways in which each corner can be oriented (without changing position of the corner) = 3⁷                                                                        (since, orientation of 8^th corner depends on the orientation of first 7 corners)

• Number of ways in which all the edges can be permuted (without changing orientation) = 12!/2

    (since, even permutation of corners implies even    permutation of edges)

 

• Number of ways in which all the edges can be oriented (without changing position of the edge) = 2¹¹ (since, orientation of last edge depends on the orientation of the first 11 edges)

Therefor, total possible states of the 3x3x3 Rubik's Cube are:

8! x 37 x 12!/2 x 211 = 43, 252, 003, 274, 489, 856, 000

Generator of G: { D, U, B, F, R , L } can generate any set of moves.

Rubik's Cube group:

Let, G = set of all possible cube moves in a 3x3 Rubik's Cube

* = operation which concatenates two sets of cube moves

Here, * is binary operation because it can be defined as a function:

* : GXG -------> G

Since, G is non-empty and * is a binary operation, therefore (G,*) is called Rubik's Cube Group because it satisfies the following properties which are necessary for a group:

a)Associativity : Since, concatenation of moves can be treated as function-composition. So, concatenation is associative because function composition is always associative.

b)Closure : Concatenation of any two sets of moves from G belongs to G.

c)Inverse: Since, for any set of moves, we can apply the moves in reverse way.

d)Identity element: Let E be the empty set of moves which means that E can be any set of moves which does not change the state of the cube.

For example, E can be RRRR or LLLL or U2U2 etc.

Since,

E * X = X*E = X

So, E is called the identity element of (G, *)

(G,*) is non-abelian: 

• A group is said to be abelian if every two elements of that group commute.

• (G, *) is non-abelian because UR is not same as RU.

Since, orientation of the 6 centre stickers is always fixed and also there are total 6x9 = 54 stickers, therefore, only 48 stickers needs to be solved.

So, G can be considered as a subgroup of S₄₈ .
Since, solving a cube requires only permutation and orientation of stickers. Therefore, we consider two subgroup C_o and C_p .

where, 

C_o = group which involves only orientation of blocks and not permutation
 
C_p = group which involves only permutation of blocks and not orientation

C_o is normal subgroup of G:

• A group N is said to be normal if gNg^-1 belongs to N for all g belong to G.

For example:

1) 
 

B R' D2 R B' U2 B R' D2 R B' U2

twists two opposite corners of the top face

2) 
 

R U D B2 U2 B' U B U B2 D' R' U

flips back edge of top face and right edge of top face

Now, we can check normality of C_o by applying moves g = RU (to keep things simple , we can use g = RUFFDLBU...) . Therefore, g^-1 = U^-1 R'. Now, if we apply g, then apply (1) and then apply g^-1 , we will get the from two corners get twisted. We will find that they gets oriented without changing there position means Without being permuted. Therefor, this new set of moves g*(1)*g^-1 belongs to C_o . You can check that, this is valid for any value of g.

Hence, C_o is normal. 

C _p: The elements of this group permutes the edges or corners or both without changing their orientation.

For example:

C_p = {UU, DD, F, B, LL, RR, RR U' F B' RR F' B U' RR}

RR U' F B' RR F' B U' RR :: permutes the three edges other than the left edge of the top face in anti-clockwise direction.

Note: Intersection of C_o and C_p is identity and G = C_o X C_{p .}

C_o is Abelian Group: Since, the piece keeps oriented at their own position as the moves are applied from the set C_o .

Co = Z2^11 x Z3^7

C_p : It consists of permutations of corners and edges without changing their orientations.It has two normal subgroups A₈ and A₁₂ .

 
where, A₈ and A₁₂ are even permutations in which we can swap two blocks . We can also take a permutation which swaps two corners and two edges

Therefore,

Cp = (A8 x A12 ) x Z2

Hence, G is isomorphic to Z2^11 x Z3^7 x (A8 x A12 ) x Z2

Interesting Facts: 

• 1260 is the largest order of an element in G.Example: (R U² D^-1 B D^-1)¹²⁶⁰ = E 

• Any scramble 3x3x3 Rubik's Cube can be solved in at most 20 moves. This is known as God's Number.

• Starting from the solved state of cube, if the same move-sequence S is applied successively, then the cube will return to the solved state again.