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# Rubik's Cube theory

## Group theory

Rubik's cube can be viewed as a mathematical group, where each element of the group is a permutation. As a group, it has the following properties:

Closure
If P1 and P2 are two permutations in the group, then P1P2 (i.e. the result of P1 followed by P2) is also a permutation in the same group
Associativity
Performing P1 followed by P2P3 is the same as performing P1P2 followed by P3.
Identity
There is a permutation in the group in which no pieces are moved.
Inverse
For each permutation in the group, there exists an inverse permutation which has the reverse effect.

Rubik's Cube also has a number of subgroups, each having these same 4 properties.

The property of groups that is perhaps most interesting to cubists is closure, which means that each operation within a particular group will take you to another element in the same group. This means that if you choose a subgroup and use only operations from that subgroup, then your cube will remain within that subgroup.

### #1

Suppose you restrict yourself to the subgroup that is generated by the set of operations consisting of 90 degree turns of the top and bottom faces and 180 degree turns of any of the side faces. In this subgroup, each operation preserves the top/bottom stickers so that they are always facing either up or down.

 In the example to the left, clicking play will always keep all yellow/white stickers on the top and bottom.

### #2

An interesting group that we make use of in the Heise method is the group generated by 180 degree turns of the front and back sides, and 90 degree turns of any of the other sides.

 Using the earlier frame of reference for edge orientation, it is easy to see that all permutations in this group preserve edge orientation.

This knowledge is very useful since by the second half of the Heise method we have already oriented the edges and need to be careful after that point to preserve their orientation. By restricting yourself to only 180 degree turns of the front and back sides and 90 degree turns of any of the other sides, this orientation can be easily preserved.