U (F R) means do parenthesis first, so do F then R then do U, clearly non-associative.

This caused my best laugh today. I have this theory that your "BS" in math doesn't stand for bachelor of science.

Perhaps I am not very good at making clear explanations, but that is no reason to throw out insults.

Let me explain this in as simple terms as I can:

First a set is defined. This can be any list of objects, preferably with something in common.

Then an operation is defined. This can be anything from simple addition to the assembly of a computer.

Once these are established it is possible to begin to determine if the set with the defined operation satisfies the requirements of a group.

Associativity is defined, as I stated earlier, (X*Y)*Z = X*(Y*Z). A set, with the defined operation, is considered associative if it satisfies the above for ALL X, Y, Z in the set.

Example: addition is associative, (5+2)+3 = 5+(2+3) True

Example: division is NON-associative, (6/3)/2 = 6/(3/2) false. For the Left hand side, 6/3 is 2. 2/2 is one. For the right, 6 is to be divided by 3/2, or 1.5, yielding 4.

For the cube, it becomes difficult to define the specific operation using face turns, simply because group theory does not formally allow different operations such as F, R, and U, or even +, -, x. (this is yet another reason it is difficult to apply group theory to a cube). I am uncertain how to get around this, but assuming it was figured out, the face turns are still non associative.