Group Structure
The following uses the notation described in How to solve the Rubik's Cube. The orientation of the six centre facets is fixed.
G can be defined as the subgroup of the full symmetric group S48 generated by the 6 face rotations.
We consider two subgroups of G: First the group of cube orientations, Co, which leaves every block fixed, but can change its orientation. This group is a normal subgroup of G. It can be represented as the normal closure of some move that flip a few edges or twist a few corners. For example, it is the normal closure of the following two moves:
- (twist two corners)
- (flip two edges).
For the second group we take G permutations, Cp, which can move the blocks around, but leave the orientation fixed. For this subgroup there are more choices, depending on the precise way you fix the orientation. One choice is the following group, given by generators (the last generator is a 3 cycle on the edges):
Since Co is a normal subgroup, the intersection of Co and Cp is the identity, and their product is the whole cube group, it follows that the cube group G is the semi-direct product of these two groups. That is
(For technical reasons, the above analysis is not complete. However, the possible permutations of the cubes, even when ignoring the orientations of the said cubes, is at the same time no bigger than Cp and at least as big as Cp, and this means that the cube group is the semi-direct product given above.)
Next we can take a closer look at these two groups. Co is an abelian group, it is
Cube permutations, Cp, is a little more complicated. It has the following two normal subgroups, the group of even permutations on the corners A8 and the group of even permutations on the edges A12. Complementary to these two groups we can take a permutation that swaps two corners and swaps two edges. We obtain that
Putting all the pieces together we get that the cube group is isomorphic to
This group can also be described as the subdirect product, in the notation of Griess.
Read more about this topic: Rubik's Cube Group
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