I am looking for the order of the stabilizer group of $D_{10}$. I know that ${G_x} = \{g \in G : gx = x\}$. I am curious what to use for $x$ though? Should I just cycle through elements of $D_{10}$ and see how many $x$'s are fixed by each element of $D_{10}$?
My approach so far (trying the above) is:
For identity element, 10 vertices are all fixed.
For any rotation, no vertices are fixed.
For any reflection, similarly, no vertices are fixed (draw out a 10-gon to see this).
Is this right? I know that the order of the stabilizer of $D_{10}$ is not 10, so I'm sure I made a mistake somewhere. Did I not consider enough points?
Thanks.