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Following the discussion about the [A | B] notation:

First, I note that the commutator notation is a standard mathematical notation having nothing really to do with cubing. If you view Singmaster's notation as basically a use of standard mathematical notation, then use of the commutator notation ( [P,Q] ) comes "for free" so to speak. Unfortunately (in my view), the cubing community has extended Singmaster's notation in a number of ways that go against its mathematical origin. I tend to dislike such extensions, especially when cubing notation extensions actually conflict with mathematical notation.

In particular I would argue that the notation:

(A B)*1.5

or

\( (A B)^{1.5} \)
(corresponding to the repetition notation Singmaster used)

is especially bad as it doesn't really make any sense mathematically if A and B are taken to be group elements, and certainly wouldn't mean (mathematically speaking) the same thing as A B A.

(I remind everybody that in Singmaster's notation, repetition was represented by exponentiation, since exponentiation corresponds to repeated multiplication for a multiplicative group.)

Bruce: if you were to have a notation system for commutators and conjugates (and A B A, in addition) what system would you prefer as to not contradict the pure mathematical notation developed by Singmaster?

You criticize our current notation system, and suggested ones, but do not provide an alternative.
Perhaps people would be more keen to stray from disregarding the notation's mathematical origin if a better or equally understandable notation was suggested?

Unfortunately (in my view), the cubing community has extended Singmaster's notation in a number of ways that go against its mathematical origin. I tend to dislike such extensions, especially when cubing notation extensions actually conflict with mathematical notation.

I see where you're coming from, but Singmaster's notation is certainly not ideal, for a few reasons. There's convenience (it's a hassle to have to use superscripts whenever we want to describe repeated turns, and the ^-1 notation is awkward if you use it a lot) and there's the need for notation that doesn't exist in group theory (or, if it does exist, is very awkward). The truth is, normal group theory notation is just not adapted for writing long sequences of elements in a way that is concise and easy to remember. (It IS useful for formal manipulation, when there are many possible types of operation that need to be carefully distinguished.) Rather than trying to conform to what's useful in some completely different field, we should be trying to change our current notation to make it as useful as possible for this field.

Stachu: I looked at your examples and I just don't see A B A being a natural form. It certainly does appear sometimes, but there's never an intuitive reason that the alg has to look that way (unless A is self-inverse). I see it as just an aid to memorization. Perhaps you would be interested in the idea of color-coding groups of moves? For instance, something like RUR' could be blue, something like R'FRF' could be red, any 2+ move sequence repeated inside one alg could be green, and any 2+ move sequence repeated (in inverted form) inside one alg could be yellow.

Bruce: if you were to have a notation system for commutators and conjugates (and A B A, in addition) what system would you prefer as to not contradict the pure mathematical notation developed by Singmaster?

You criticize our current notation system, and suggested ones, but do not provide an alternative.
Perhaps people would be more keen to stray from disregarding the notation's mathematical origin if a better or equally understandable notation was suggested?

Well, like some others, I'm not convinced of the the need for a "PQP" notation. (And I also strongly dislike when people use "B" as both a constant and a variable at the same time!) There are standard mathematical conventions for both commutators and conjugation. Exponentiation is what's used in mathematics for conjugation. This does not conflict with exponentiation for repetition because repetition has a numerical exponent, while for conjugation the exponent is a group element.

There is an issue in mathematics that the notation is used differently in different places. [P,Q] can mean \( PQP^{-1}Q^{-1} \) or \( P^{-1}Q^{-1}PQ \). Similarly, \( P^Q \) can mean \( Q^{-1}PQ \) or \( QPQ^{-1} \). The latter conventions in each case seem to make more sense for cubing and, of course, the latter interpretation of commutator notation is what we actually use in the cubing community. The conjugation notation does not seem to be well received in the cubing community, though.

The nice thing about simply regarding the notation as mathematical notation is all the notation that comes for free from mathematics. For example, generator notation (e.g. <R,U> ) is standard mathmematical notation. When we regard the notation as mathematical notation, we don't need to *define* these things; they've been defined for us.

I acknowledge there's also some issues with big cubes and other puzzles where the where it's been awkward to maintain compatibility with mathematical notation conventions.

I see where you're coming from, but Singmaster's notation is certainly not ideal, for a few reasons. There's convenience (it's a hassle to have to use superscripts whenever we want to describe repeated turns, and the ^-1 notation is awkward if you use it a lot) and there's the need for notation that doesn't exist in group theory (or, if it does exist, is very awkward).

I am not advocating to actually use superscripts, especially when it's awkward to do so. And Singmaster actually used the apostrophe instead of an exponent of -1, even though that's not standard mathematical convention. Myself, I prefer to omit the * and write, say, (FR'F'R)3 instead of (FR'F'R)*3 (or more logical mathematically speaking, (FR'F'R)^3), consistent with writing U2 rather than U*2 or U^2. It's more conventions like (R) instead of x that I don't like as it unnecessarily gives parentheses a new meaning inconsistent with it's usual meaning of grouping. (Yeah, I know a lot of cubers probably dislike x, y, and z, or at least the way they're mapped to the axes, but I'd rather use something like Rc (like we have Rw) than (R). But I'm also OK with x, y, and z as they are.)

There is an issue in mathematics that the notation is used differently in different places. [P,Q] can mean \( PQP^{-1}Q^{-1} \) or \( P^{-1}Q^{-1}PQ \). Similarly, \( P^Q \) can mean \( Q^{-1}PQ \) or \( QPQ^{-1} \). The latter conventions in each case seem to make more sense for cubing and, of course, the latter interpretation of commutator notation is what we actually use in the cubing community.

Hang on. First, neither notation makes "more sense" for cubing (either would be equally valid IMO, but I personally prefer the ones that put the non-inverted things first); second, I'm pretty sure [P,Q] is generally written to mean PQP'Q'. There is an issue in group theory where it isn't always entirely obvious what direction the multiplication goes; fortunately it's always left-to-right in cubing (glad we didn't take the right-to-left order from math ).

Unfortunately, few cubers are serious mathematicians, so this isn't anywhere near as useful as it sounds. And whenever we can invent a notation which is better for our purposes than the one math uses, I say we should use it.

Myself, I prefer to omit the * and write, say, (FR'F'R)3 instead of (FR'F'R)*3 (or more logical mathematically speaking, (FR'F'R)^3), consistent with writing U2 rather than U*2 or U^2.

Right. Personally I write P^3 and P3 only (never P*3) but P3 is definitely not a standard mathematical notation, despite how convenient it is. To be consistent, you'd have to use a superscript.

It's more conventions like (R) instead of x that I don't like as it unnecessarily gives parentheses a new meaning inconsistent with it's usual meaning of grouping.

I don't like this either (for the same reason as the conjugate notation). Just like conjugate notation, though, it looks like not everyone uses it - so that's nice.

Huh? I was talking about the consistency (or inconsistency) between the syntax used for repetition of a standard generator and the syntax used for repetition of a sequence of moves. You don't have to use superscripting in order to be consistent between the two. You seem to be talking about consistency with mathematical notation here.

We already do write (FR'F'R)3 (well, most of us). But your point at the beginning was that we should try to use standard mathematical notation whenever possible. Do you no longer agree with that?

Well, of course I acknowledge there is some awkwardness in superscript notation, and I don't expect the speedcubing community to change to using it on a wide scale. Can we even do general superscripts on this forum without the math tag? (I know there is a superscript 2 character in UNICODE that is one workaround, but superscripts are still awkward to use and also often results in text needing more space.)

Did I say "whenever possible"? I think that wording may be a little stronger than what I was trying to argue. But I think if the notation departs too much from mathematical conventions, then it becomes more awkward to use the notation to express the math concepts and other related math notation (like "<R,U>" for expressing group generators).

U2 is a single move and there's no need for U*2. That'd be like using U*1 for U. Unnecessary because it's not a combination of moves whereas triggers like FR'F'R, and RUR'U' are series of moves. When executing a series of moves like that it's just easier to put *2/*3 than typing it out three times because it's shorter and simpler. You have to tell the person how many times to execute a trigger but when doing a U2 it's only ever done once in an algorithm.

You can regard U2 as a single move but it still means the move U repeated two times. Singmaster did not *define* a move "U2". He defined the move \( U \) as an element of the Rubik's Cube group, and let mathematical conventions "define" what \( U^2 \) means.

Don't be silly. Using a repetition count of 1 is just as unnecessary for a standard generator as it is for a sequence like (F R' F' R). That analogy doesn't work at all.

In Singmaster's notation, \( U \) is a group element. Writing \( U^2 \) means doing \( U \) twice. \( (F R' F' R) \) is another group element. \( (F R' F' R)^2 \) means doing it twice. Singmaster used the exact same notation for repeating a sequence as he did for repeating a standard generator such as \( U \). He didn't even have to define what exponentiation means. What raising a group element to a power was defined long before Rubik's Cube was even invented.

I might add that mathematicians had also (a long time ago) defined what it means to raise a group element to a negative power. And specifically, a group element raised to a power of -1 is always the inverse of that group element. Singmaster defined the apostrophe symbol (although we typically say "prime") as a shorthand for raising something to the power of -1 (in other words, its inverse). So by defining the "prime" operation along with six basic moves {U, D, L, R, F, and B} as group elements, mathematics conventions provided notation for describing not only all 18 face turns, but all 43+ quintillion elements of the Rubik's Cube group.

Unnecessary because it's not a combination of moves whereas triggers like FR'F'R, and RUR'U' are series of moves. When executing a series of moves like that it's just easier to put *2/*3 than typing it out three times because it's shorter and simpler. You have to tell the person how many times to execute a trigger but when doing a U2 it's only ever done once in an algorithm.

Since U and (F R' F' R) are both group elements, we don't need two different notations for what it means for doing U twice and doing (F R' F' R) twice. The * in (F R' F' R)*2 is just as unnecessary as the * in U*2.