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Posts Tagged ‘torsors’

MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but

  • I happen to be a member of the Rubik’s Cube club here, and
  • Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.

Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?

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Now that we’ve discussed group actions a bit, it’s time to characterize them. In this post I’d like to take a leaf from Tim Gowers’ book and try to make each step taken in the post “obvious.” While the content of the proofs is not too difficult, its motivation is rarely discussed.

First, it’s important to note that there is a way to take direct sums or disjoint unions (category theorists would say coproducts) of group actions: given a group G acting on two sets S, T, one defines an action on their disjoint union S \sqcup T in the obvious way: pick one action or the other. (Disjoint unions differ from unions in the usual sense because we relabel the elements of the sets so that they cannot intersect.) There’s a great reason to do this, and cycle decomposition showed us a special case: every group action is a direct sum or disjoint union of the action on its orbits.

This is the first step toward a structure theorem. Since the group action cannot “mix” between two orbits, it acts “independently” on orbits, and any question we might want to ask about the group action can be answered by looking at each orbit separately. A group action with a single orbit is called transitive, which means that for every x, y in the underlying set there exists G such that gx = y. So to classify group actions it suffices to classify transitive group actions.

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I find non-canonical isomorphisms very interesting, but I wish I knew more examples. To be vague, an isomorphism (perhaps in a category) is said to be non-canonical if it requires making an “arbitrary choice.” One of the reasons I find them interesting is that we often think of objects only up to isomorphism, but in order for certain things to make more sense we must distinguish between objects that are non-canonically isomorphic. Here are the examples I know of.

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