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Can someone explain how these answers were derived? I have trouble with understanding the concepts of doing this sort of problem. I know the stabilizer is that which brings the element to itself. I am confused on what the orbit is ($\text{orb}_G (i)=\{\alpha(i)|\alpha\in G\}$).

rover2
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  • You have a group $G$ and a set $S$. Let $i \in S$. The orbit of $i$ with respect to $G$ is the subset of elements in $S$ that can be reached by applying a member of $G$ to $i$. – FullofDill Nov 28 '17 at 15:58
  • are you able to explain how the stabilizer and orbit of the elements above are obtained? i am not seeing it – rover2 Nov 28 '17 at 16:00
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    https://math.stackexchange.com/questions/253179/intuitive-definitions-of-the-orbit-and-the-stabilizer ? How can you know the definitions of orbit and stabilizer and yet not be able to identify their elements? It's a little contradictory... – rschwieb Nov 28 '17 at 16:00

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Given a set of symmetries $G$ acting on a set $X$ of symbols, one of which is $x$:

To find the stabilizer of $x$: for all $g\in G$, compute $g\cdot x$. If $g\cdot x=x$, write down $g$. When you are done, you have the stablizer of $x$.

To find the orbit of $x$: for all $g\in G$, compute $g\cdot x$. Write down the resulting element of $X$ if you have not seen it previously. At the end, you have the orbit of $x$.

What I have written here is nothing more than a verbose regurgitation of the definition of the two sets.

rschwieb
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