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October 6

Definition of "group"

I'm struggling to reconcile two statements in the group (mathematics) article. The lead says the group operator must satisfy the group axioms, namely closure, associativity, identity and invertibility. The lead image caption says The manipulations of this Rubik's Cube form the Rubik's Cube group. However, the Rubik's cube operations are clearly not associative, at least as I understand the term. For instance LFU would be evaluated as (LF)U by the usual order of operations rules. L(FU), on the other hand is evaluated in the order FUL by the same rules. Patently FUL != LFU. SpinningSpark 11:16, 6 October 2018 (UTC)[reply]

I'm not sure why you want to reverse the order of the terms here. That doesn't happen though, and this really is associative. More concretely, the elements of the Rubik's cube group are permutations of the cube's facets. That is, they're functions from the set of facets to itself, and the group operation is then just ordinary function composition, which is certainly associative. –Deacon Vorbis (carbon • videos) 11:48, 6 October 2018 (UTC)[reply]

Hmm, maybe part of the confusion might be stemming from the fact that function composition is normally written right-to-left, while cube moves are written left-to-right? –Deacon Vorbis (carbon • videos) 11:58, 6 October 2018 (UTC)[reply]

The property that allows you to swap the order of operations from LFU to FUL is Commutativity, which is a property that the Rubik's Cube group does not have. Iffy★Chat -- 11:52, 6 October 2018 (UTC)[reply]

Exactly so. But the associativity property F(UL) implies I should evaluate the result of UL first before evaluating the result of F. That results in the operations carried out in the order ULF which in turn requires the operators to be commutative if it is to be equal to FUL. Clearly, that isn't so. SpinningSpark 12:07, 6 October 2018 (UTC)[reply]

Yes, you need to evaluate UL first; call the result X. The expression then asks for FX, while you're trying to change it to XF, which is something different. –Deacon Vorbis (carbon • videos) 12:26, 6 October 2018 (UTC)[reply]

Well, not "first" in the sense that if this was written down, you'd physically perform the moves first. "First" here just means that you think of grouping these two moves together as a single move (which I called X above). But you'd still have to perform the X move after F. Maybe that's the issue here. –Deacon Vorbis (carbon • videos) 12:43, 6 October 2018 (UTC)[reply]

The root problem here is that standard mathematical notation is backwards and this results in people getting confused. If you want to evaluate ln(cos(.5)) on a calculator, first you enter .5, then you press the cos key, then you press the ln key, so really you're reading the expression right to left. Not that there's anything wrong with reading right to left; there are plenty of written languages where it works just fine, but in English people read left to right, so it'd odd and confusing to have mathematical expression reading right to left embedded in English text. When I do computations involving a lot of function compositions I'll often use exponential notation for functions, i.e. x^f instead of f(x), that way expression reads in the correct order; But if calculations need to be shown to anyone else I then need to reverse everything to conform to the standard; whether it's confusing or not, there is too much inertia for writing functions on the left for it to change. In any case, whether you write operations from left to right or right to left, as long as you're consistent the associative law will hold. --RDBury (talk) 13:53, 6 October 2018 (UTC)[reply]

The group operation in Rubik's Cube group is to make one permutation followed by another, where a permutation is any possible result of a sequence of rotations. Think of (LF)U as: First do a compound move consisting of L followed by F. Then do U. L(FU) means: First do L. Then do a compound move consisting of F followed by U. Both cases correspond to just doing L, F, U in that order. Nothing is swapped. PrimeHunter (talk) 14:15, 6 October 2018 (UTC)[reply]