Finite Group G-set Stabiliser and Orbit: Orbits and Stabilisers Homework

[Ted123]

Homework Statement

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

http://img36.imageshack.us/img36/1962/grouplg.jpg

Homework Equations

The stabiliser of $x$ is defined: $$\text{stab}_G (x) = \{ g\in G : gx=x \}$$

The Attempt at a Solution

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

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

 

[jbunniii]

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

Yes. Your answer looks fine to me. The question also asks for the stabilizer of x. Did you answer that part? (I assume the answer is pretty obvious.)

 

[Ted123]

Yes. Your answer looks fine to me. The question also asks for the stabilizer of x. Did you answer that part? (I assume the answer is pretty obvious.)

OK good. I think the stabiliser is just $$\text{stab}_G (x) = \left \{ <br /> \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right \}$$

 

[jbunniii]

OK good. I think the stabiliser is just $$\text{stab}_G (x) = \left \{ <br /> \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right \}$$

Yep.