Rubik's Cube 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.

#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.

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.