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Consider the action of the full symmetric group $S_3$ on the cube $[0,2] \times [0,2] \times [0,2]$.

Classify the orbits of this action and determine their cardinalities.

My Answer: What I note is that the orbit can have six possibilities: $(j,i,k)$, $(k,i,j)$, $(i,j,k)$, $(j,k,i)$, $(k,i,j)$; of the form $(i,i,k)$ with stabilizer $2$ and orbit $3$; or of the form $(i,i,i)$ just a stabilizer.

So does this mean that the orbits of the action we're looking at can be $(0,2,0)$, $(0,0,2)$, $(2,0,0)$; and $(2,2,0)$ or $(0,2,2)$ or $(2,0,2)$?

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2 Answers

Hint: You correctly determined the orbits of the elements $(0,2,0)$ and $(2,2,0)$. But notice that the interval $[0,2]$ has more elements than just $0$ and $2$. Therefore, most orbits will have six elements. For instance, consider the orbit of the element $(0,1,2)$. It consists of the elements $$ (0,1,2), (0,2,1), (1,0,2), (1,2,0), (2,0,1), (2,1,0). $$ Finally, the orbit of the element $(1,1,1)$ consists of only one element.

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How does $S_3$ act on the cube? By permutation of coordinates? Also, your terminology is slightly difficult to follow, even though I believe you mean the right thing. Your last line doesn't quite make sense, though. Why are you only looking at elements with entries 0 and 2?

But in principle you are right. If we are talking about the same action, then there are three different cases:

  1. Orbits of elements with all three coordinates the same, these orbits are singletons.

  2. Orbits of elements with exactly two coordinates the same, these are of length 3.

  3. Orbits of elements with all three coordinates different. These are of length 6.

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It acts on by permutation of coordinates. –  Buddy Holly Jan 24 '12 at 9:08
    
How did you get that if orbits of elements with 2 coordinates the same, they are of length 2, is it just 3 choose 2? with 2 of them the same –  Buddy Holly Jan 24 '12 at 9:10
    
can you classify the orbits? The cardinalities make sense. To my understanding, for the orbits we need to list the coordinates, but it looks like I'm not getting that –  Buddy Holly Jan 24 '12 at 10:23
    
Sorry, if two coordinates are the same, the size of the orbit is 3. I corrected this. –  Stefan Geschke Jan 24 '12 at 22:11
    
I would say there are exactly three different types of orbit, the ones mentioned in my answer. Do you need something more? –  Stefan Geschke Jan 24 '12 at 22:16
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