Symmetric Group S3 Symmetry 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 $(2,0,2)$?
 A: 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.
A: 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:


*

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

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

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