Find this on YouTube:
I know cubing is somehow related to group theory and some mathematical stuffs, but I don’t actually know anything about it.
Where should I start if I want to learn more about this?
(I almost forget what Abelian group is and my major is applied mathematics lol)
I'm not entirely sure what the exact thing is you want explained, but I'll give you a definition of what a (non-)abelian group is, and an explanation as to why cubes are groups, and why the cubes Chris Choi solved in this video are not non-abelian, in case that helps.
Basically, a set G is called a group under an operation · (which is a function from GxG to G) if the follow things are true:
1. There exists and element e in G such that for all g in G, e·g=g·e=g (i.e. a neutral element exists)
2. For all g in G, there exists and element g^-1 in G such that g·g^-1 = g^-1·g = e (i.e. every element has an inverse)
3. · is associative, that is, for all f,g,h in G, f·(g·h)=(f·g)·h
(I tried to do this in LaTeX, but the preview option did not show any conversion to symbols, so I suppose we're stuck with this for now)
A group is abelian if for all g,h in G, we have g·h=h·g, and non-abelian else.
Now, a 3x3, for example, is not a group in and of itself, but rather the set of 3x3 states are a group under concatenation. That is, say you have two states, you can find two sequences of moves that lead you to these states, concatenate them, and see where that sequence leads you. The neutral element, of course, is the solved state, the inverse to a state s is simply the state where applying a solution for s to a solved cube leads you (that's the point of something being a solution anyway). Concatenation is associative by nature, so it all works out. The 3x3 is a non-abelian group, because the order matters: take two solved cubes, do a T perm and then a sune on one of them, and on the other do the sune first and then the T perm. These lead to different states.
For clock, the order in which you do the moves does not matter, hence it's and abelian group. SQ-1 is obviously not a group, and I guess you could make 4x4-7x7 into non-abelian groups by defining some sort of distinction between currently identical center pieces, but that's just not how these cubes are treated. (For example, there are many solved 4x4 states. Which one do we consider the neutral element? Certainly it must be unique, because if there are two (say e and e'), then e·e' = e and e·e' = e', so e=e').
