Orbits and Stabilisers

1. Jan 31, 2012

Ted123

1. The problem statement, all variables and given/known data

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 [Broken]

2. Relevant equations

The stabiliser of $x$ is defined: $$\text{stab}_G (x) = \{ g\in G : gx=x \}$$
3. 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?

Last edited by a moderator: May 5, 2017
2. Jan 31, 2012

jbunniii

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.)

3. Jan 31, 2012

Ted123

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

4. Jan 31, 2012

Yep.