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.)
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