Inverse element of dihedral group... I am having a hard time conceptualizing what is the inverse element of, for example, a dihedral group on the equilateral triangle. Can anyone explain this to me?

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. 
