Inverses of dihedral group elements

Question

I am studying group theory and I'm having a hard time conceptualizing what the inverse element of an element of the dihedral would be like, on the equilateral triangle, for example. Any explanation is very well received.

Answer 1

It may help to see all of the symmetries of an equilateral triangle, and then writing down the subsequent rotations or reflections required to get back to the original image. For the equilateral triangle, its symmetry group is $$ \{e, r, r^2, s, rs, r^2s\} $$ where $r$ is a counterclockwise rotation of 120$^\circ$ and $s$ is a reflection across an axis of symmetry (there are other ways to define its dihedral group; this is just one of them). Visually, For example; looking at $r^2s$, we can get it back to the original image by rotating it once, and then reflecting it across the axis of symmetry. See if you buy that first. If you do, then we can say the inverse of $r^2s$ is $sr$ (rotation first, reflection second), which makes sense, because $$ (r^2s)(sr) = r^2s^2r = r^2r = e $$ since $r^3 = e$ and $s^2 = e$. So try to think about the inverses visually, then write them down and check if it makes sense algebraically.