Stabiliser Groups of a vertex/edge of a square

[Ratpigeon]

Homework Statement

Given the dihedral group of symmetries of a square; what is the stabiliser group of a vertex (or edge)?

Homework Equations

The stabiliser group is G_x={g$\in$G|gx=x}
I guess for a vertex/edge that means the transformations in D4 (generated by reflection in x-axis and rotation by 90°)

The Attempt at a Solution

Doodling in my notebook, I have determined that the only way that the order of vertices/edges can be changed is by reversing them - which means that the only unique elements of G that map a vertex/edge to itself is the identity, and one other - which depends on the vertex/edge, but is basically a rotation combined with a reflection, that reverses the order of the vertices and then rotates it back to the relevant spot... are theere any others? and how can I do it algebraically to demonstrate a complete stabilising subgroup?

 

[kurt.physics]

The stabiliser group of a vertex/edge in the dihedral group of symmetries of a square is the subgroup generated by the identity and the reflection in the axis that passes through the vertex/edge, followed by a rotation of 180°. This subgroup has order 2.