## A mathematical view of Coordinates: Cosets |
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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. The following table describes the subgroups H, which correspond to the various types of coordinates used in Cube Explorer.
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. |
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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. |