(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

For the finite group [itex]G[/itex] and G-set [itex]X[/itex] below, find the stabiliser [itex]\text{stab}_G(x)[/itex] of the given element [itex]x \in X[/itex] and describe the G-orbit of [itex]x[/itex].

http://img36.imageshack.us/img36/1962/grouplg.jpg [Broken]

2. Relevant equations

The stabiliser of [itex]x[/itex] is defined: [tex]\text{stab}_G (x) = \{ g\in G : gx=x \}[/tex]

3. The attempt at a solution

I get: [tex]\text{orb}_G \left ( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right ) = \left \{ g \begin{bmatrix} 1 \\ 0 \end{bmatrix} : g\in G \right \}[/tex] [tex]= \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} \frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{bmatrix}, \begin{bmatrix} -\frac{1}{2} \\ \frac{\sqrt{3}}{2} \end{bmatrix} , \begin{bmatrix} -1 \\ 0 \end{bmatrix} , \begin{bmatrix} -\frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix} , \begin{bmatrix} \frac{1}{2} \\ -\frac{\sqrt{3}}{2} \end{bmatrix} \right \}[/tex]

But when the question says 'describe the G-orbit of [itex]x[/itex]' does this mean 'find [itex]\text{orb}_G (x)[/itex]' or what?

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# Orbits and Stabilisers

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