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For my abstract algebra course this quarter, I tried to write a paper that would cover the basic ideas of group theory as they applied to the Rubik's Cube – without assuming too much familiarity with terminology. It's intended to be accessible to members of this forum. If you've been curious about this "group theory" thing, but didn't know where to start, you might want to take a look.

The examples toward the end of the paper aren't well-developed yet, but the rest should be mostly in order. If you decide to read this, I'd appreciate any feedback on how to make it more relevant to cubers (or mathematicians who might not be cubers).
If you haven't learned group theory, did it make you understand something new, or is it just confusing?

First, I note one error where you attribute the finding that approximately 50% of random pairs of cube group elements generate the group as being due to Rockicki in 2010, when it was actually due to Schoenert in 1995. (Reread the thread you mention, particular post #13.)

On page 2, you define what raising a group element to a (positive) power means, and what raising a cube group element to a power of -1 means. On page 3. Once you describe what \( R \) means, it is clear from the notation defined on page 2 that \( R^2 \) would mean, and what \( R^{-1} \) would mean. Then you can say that that in the cubing community, that these are often simply notated as "R2" and "R'" for convenience (and note that the apostrophe or prime symbol is really something Singmaster introduced as a shorthand for writing an exponent of -1). I find it surprising that since you already introduced the mathematical notation on the previous page, you aren't relating the popular notation conventions of the cubing community today to the mathematical notation you already talked about on the previous page.

Your paper seems to primarily talk about cube group elements as if they represent "states," while I prefer to think of them as "transformations." When one talks about composing group elements, you ultimately need to describe the group elements in terms of transformations, so you can define the composition of two group elements in terms of the composition of two transformations. (If I roll a die a couple of times, I might describe the resulting states as, for example, "3" and "6". But what state do I get if compose "3" with "6"? This is to illustrate the problem of using states to define group elements.)

Yes, your paper does get into describing permutation of stickers and "group action." I guess what bothers me is that instead of saying a cube group element really represents a permutation of the stickers, it says a state represents a permutation (that is, a transformation, rather than something static). So you are now using the term "state" to represent a transformation, rather than a "position," which I view as something that is static like the "3" and "6" in my die example.

Once a cube group element is seen as a transformation (dynamic, so to speak) rather than a static position, we know longer see a problem with the 4x4x4 cube group not being the same size as the number of (static) positions of the 4x4x4 (non-supercube) cube. We view the 4x4x4 cube group as the number of possible transformations on that cube (the number of different ways stickers can be moved around) instead of the number of static states of that cube. The digression with the 4x4x4 cube on page 4 seems to me somewhat awkward in that it brings up yet another group, additional move types, and the concept of supercube without really explaining what a supercube is, etc. The lack of the 4x4x4 having fixed centers to use a reference is also not mentioned.

I have no previous background with group theory. After reading your article though I think I understand group theory's basics fairly well. Some sections of it were a bit confusing though. I had to read through 2.1 a couple times to get an idea of it. Thank you for sharing this, it was quite enlightening.