A mathematical view of Coordinates: Cosets

If you have a group G and a subgroup H, then for each g from G the set {a*g | a from H} is called a right coset of H.

Each scrambled cube can be seen as a permutation with attached orientations. All these permutations define the group G.
Every coordinate-value used in Cube Explorer can be mapped onto a right coset, where the subgroup H mentioned above is defined by the type of the coordinate.

The following table describes the subgroups H, which correspond to the various types of coordinates used in Cube Explorer.

subgroup of G	coset coordinate	used in
all permutations with corner orientations = 0	corner orientations coordinate range 0..2186	phase 1, optimal solvers
all permutations with edge orientations = 0	edge orientation coordinate range 0..2047	phase 1, optimal solvers
all permutations which leave the four UD-slice edges in their slice	UDSlice coordinate range 0..494	phase 1, optimal solvers
all permutations with edge orientations = 0 and which leave the four UD-slice edges in their slice.	FlipUDSlice coordinate range 0..494*2048 - 1	phase 1, standard optimal solver
all permutations which leave the corners in their place with arbitrary twist.	corner permutation coordinate range 0..40319	phase 2, optimal solvers
all permutations which leave the 8 edges of the U-face and D-face in their place with arbitrary flip	phase 2 edge permutation coordinate range 0..40319	phase 2
all permutations which leave the four UD-slice edges in their place with arbitrary flip	UDSliceSorted coordinate range 0..11879	phase 2, optimal solvers

Some of the above coordinates are "reduced by symmetries" in a second step before actually being used. We will discuss this in the next chapter.

Take for example the subgroup C0 which defines the corner orientation coordinate

C0 = { g from G with g(x).o = 0 for all corners x}

In this case the right cosets are defined by

C0*g = {a*g | a from C0}

For each element a from C0 and an element g from G and any corner x we have regarding the definition of the multiplication

(a*g)(x).o = a(g(x).c).o + g(x).o = 0 + g(x).o = g(x).o

So all elements of the coset C0*g have the same corner orientations (defined by the permutation g) and all elements of C0*g have the same corner orientation coordinate. And if on the other hand two permutations have the same corner orientation coordinate, they are in the same coset. The formal proof is a bit more lengthy:

If two permutations a and b have the same corner orientation coordinate we have a(x).o = b(x).o for all corners x.

For an arbitrary corner x set

cb:=b^-1(x).c and ob:=b^-1(x).o,
(corner x is replaced by corner cb and the orientation of cb is increased by ob when put into position x) .
We then also have

b(cb).c=x and b(cb).o= - ob
(b is the inverse of b^-1, so corner cb is replaced by corner x and the orientation of x is decrease by ob when put in position cb).

a and b are in the same coset of C0 if and only if a*b^-1 is in C0. Now we have for all corners x

(a*b^-1)(x).o = a(b^-1(x).c).o + b^-1(x).o =a(cb).o + ob = b(cb).o +ob = -ob + ob = 0 q.e.d.

So there is a one to one mapping between the corner orientation coordinate and the cosets defined by C0.