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I was reading the following paper from MIT:

The paper is not difficult to understand, it is more or less a short introduction into group theory, taking the Rubik's Cube as an example of a finite, non-abelian group.

My question: How can I use this fact to find possible ways for solving the cube, if you know what I mean? There are several ways for solving the cube, which seem to me more intuitive rather than a direct result of the group properties.

So if somebody would ask you "It is nice that Rubik's Cube can be seen as an example of a finite group, but what can I do with this knowledge? ", what would you answer?

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The key is conjugation. Knowing how to flip something here (without side effects) allows you to flip something elsewhere by conjugation ... – Hagen von Eitzen Feb 11 '13 at 14:09
Another important and useful notion of group theory is that of commutators. Almost every algorithm is built up out of simple conjugations and commutators. – Martin Brandenburg Feb 11 '13 at 14:51
would love to see these ideas developed along with their implications on the cube – user58512 Feb 11 '13 at 14:53
btw, a quick google research gives a lot of answers to the question. – Martin Brandenburg Feb 11 '13 at 14:56
At the dawn of www I wrote a page depicting key sequences (3-cycles of corner cubes, 3-cycles of edge cubes, flipping two adjacent cubes of the same type) in standard position. As anticipated by Martin, they are all constructed as commutators. As pointed out by Hagen, you should conjugate these to get similar permutations, when the participating cubes are not in standard position. The page is here, but it is in Finnish only. You have to scroll down a few pages to get to the pictures :-) – Jyrki Lahtonen Feb 11 '13 at 15:19
up vote 7 down vote accepted

From a theoretical perspective, solving Rubik's cube is a nice example of the Schreier-Sims algorithm in action, although applying the algorithm will not give in general the quickest solution.

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The book by Alexander Frey and David Singmaster, Handbook of Cubik Math, solves the cube using such group-theoretic concepts as cycles, equivalences, identities, inverses, order, commutativity, and conjugates.

There's also a book by David Joyner, Adventures in Group Theory, which has a lot of stuff about the Rubik cube, but I don't know whether it does what you want.

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