Is a dihedral group only considered for shapes that when reflected or rotated fit exactly back into place of the original image?
My confusion arises from this wikipedia article and specifically from this picture:
Are these symmetries considered a dihedral group?
On
Look, Michael, dihedral groups are easy, they are the next best thing after a cyclic group. While in a cyclic group you turn things around in a circle, see your dihedral group as a prism, like two identical cycles above each other, forming a kind of coin. Then your movements consist in turning the coin as if it were cyclic, and the extra moves being the flips of the coin and consider the "F" as a symbol stamped through the coin.
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Those four actions form a dihedral group of order $4$. They are not symmetries of the letter "F" though, they are symmetries of a line segment (2-gon). The rotations are applied to the letter "F" in the picture to help you see what they do. If they are symmetries of anything in the picture, they are symmetries of the $y$-axis (or the $x$-axis).
The dihedral group specifically refers to the symmetries of a regular polygon, so an $n$-gon under any transformation in the dihedral group $D_n$ is mapped to itself. I think the picture in this article is using an $F$ for illustrative purposes, since an $F$ has $no$ symmetry; thus you can see that switching the order of transformations really does make a difference (this would be more difficult to show on an actual square, which is symmetric, so they would need to color the vertices/edges or something like that).