I am using the following (incomplete?) definition of a symmetry:
A symmetry of a figure $F\subset\mathbb{R}^n$ is an isometry which maps $F$ to itself.
Let me illustrate my question by considering a straight line $S\subset\mathbb{R}^2$. I can immediatly identify two symmetries:
- The identity
- Rotation by $180°$
I notice that the symmetry given by a rotation by $360°$ is equal to the identity because every point $(x,y) \in \mathbb{R}^2$ is mapped to itself.
But what about the reflection about the axis coinciding with $S$? In this case every point $(x,y) \in S$ with $ x,y\in \mathbb{R} $ is mapped to itself. So the action on the straight line is exactly the same as the identity would do. Now my question: Is this counted as a seperate symmetry or as the same?
Short answer: It depends on what you want to do with the line and its symmetries. Some times you want your line to have "red and green sides", some times you don't.