Evariste Galois: Group Theory

Group theory
A major branch of modern discovered by mathematician Evariste Galois, group theory is the mathematical language of symmetry. It is the concept that is most associative with Rubik’s cube due to its invariant symmetry and the interdependence of it components. In mathematics, a “group” is a set of elements and a rule to combine those elements which satisfies certain properties. The elements of a group can consist of numbers, symmetries of a shape, or the elements themselves can be the different ways of rearranging another set of elements. (“Group”)
An important part of understanding group theory, a binary operation is rule for combining elements of a set that will consequent in another element of the same type. For example, addition

Each permutation symbolizes the result of a single element of the group. Different moves of the cube could correspond to the same final permutation and therefore to seemingly the same exact element of the group. However, a single element of the Rubik’s cube group can be expressed using different sequences of moves, just as a rotation of 4050 is the same as 450. Here, we can utilize the expression “followed by” to represent the operation in this group. Thus, one move followed by another move is itself a move of the Rubik’s cube.
The number of elements in the Rubik’s cube group is, as we calculated earlier, approximately 43 quintillion because there are many distinct permutations of the cube that can be reached by legal move (without the disassembly of the cube). Any two legal moves can be combined to result in one of those permutations.
To verify that this is a group:
- There is an identity element, namely “not performing any move”. “Not performing any move” followed by a move x equals the move

If the puzzle were commutative, it would not matter what order the move were implemented in, as long as each face was rotated an appropriate number of times. In this way, group theory provides deductive conclusions that would be applicable to all groups. Since the Rubik’s cube is a group, anything that is true for a group in general is true for the Rubik’s cube. (“Galois”) Furthermore, group theory is helpful in deducing certain algorithms. In particular, they are useful in deducing algorithms that possess a commutator structure, namely XYX-1Y-1(X and Y are moves or move sequences and X-1 and Y-1 are their inverses), or a conjugate structure, namely XYX-1 which is usually referred to as a “setup move”.(“Rubik’s Cube”) Both these concepts will be discussed