# 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..493

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 (we omit the proof here). So there is a one to one mapping between the corner orientation coordinate and the cosets defined by C0.