Hi everyone,
I am currently doing a project on the Rubik's Cube and was trying to find a clear explanation online as to what the cube group's structure was, however, I soon realised I wasn't really sure what a group structure was :S can anyone shed some light on this for me?
Cheers
strictly speaking, rubik's cube is not a group, per se, but an example of a group acting on a set. the set being of course, "the cubelets" that make up the rubik's cube, and the group being the various move sequences.
what G = Co X| Cp "means" is that we can approach the solution of the rubik's cube in 2 stages: get every cublet in the correct position, and then "fix the orientations", or (as is more often done in practice) move the cubelets into their proper positions using "orientation-preserving sequences". the latter is usually accomplished by first doing a move that "scrambles the cube somewhat", then a sequence that permutes position (3-cycles are preferred among people that solve cubes quickly), then doing the first move in reverse.
the "group structure" arises because every sequence of moves is reversible (so we have inverses), and subsequent consecutive moves can be regarded as compositions of "basic moves" (turning one face 90 degrees), which assures us of associativity (U then F then R is the same as (F then R) preceded by U). the identity is "the empty move" (doing nothing).
because of the large size of the group itself, explict description of all group elements is not feasible. so a description of the "Rubik Cube Group" is often in terms of how it can be described in terms of smaller groups (much like the way a molecule can be described by a chemical formula in terms of its atoms).
Another way of seeing that you have a group acting on a set is that when you do a move you are doing no less than permuting the cubies! Permutations=groups, QED...
A nice corollary of the group structure is that if you start with a solved cube, fix a move and apply it repeatedly to your cube then you will end up with the solved cube again. This is, as the kids say, `cool'.