I know this question may seem a bit silly and trivial, but I haven't found any precise definitions for what a "symmetry" is in my textbook for group theory or online and I feel like I don't know it rigorously enough. Specifically, the root of my question is:
Why doesn't the dihedral group of order 2n have n! elements?
For example, the symmetries of a square (that is the dihedral group of order $2n = 8$, 4 reflections and rotations) has order $8$, but why isn't, for example $(12)$ a valid element in it?
I understand that there are axes of rotations and symmetries, but is that it? Is there a more precise definition of what a "symmetry action" must constitute?