Group Axioms and Properties
The Rubik's Cube group (G, •) consists of the set of cube moves, G, and the operation of concatenation, •. Concatenation is a form of function composition.
The group axioms are:
- associativity, function composition is always associative.
- closure, the concatenation of two moves is another move.
- identity element, the empty move concatenated in either order with any other move is the same as .
- inverse element, reversing a move returns the cube to its previous position. Reversing a sequence of moves inverts the sequence.
More specifically, (G, •) is a permutation group. The basic cube moves form a generating set.
The cardinality of G is finite but large. Still, each position can be solved in 20 or fewer moves.
The largest order of an element in G is 1260. For example, .
(G, •) is non-abelian. Since is not the same as, not all cube moves are commutative.
Read more about this topic: Rubik's Cube Group
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