Just to make sure. So each loop based at a point is distinct from another loop, if it has more "winding"? Like, the loop path can pass several times through the base point p, and it's a distinct loop from a loop that only passed once over it. Pictorically, a distinct loop would be tracing the circle n times, ending at the base point. And then the inverse element would amount to "dewinding" said loop, i.e. by going in a loop in the different direction?
Is this the correct way of thinking about the fundamental group of the circle? And this can't be used on a disk, since in a disk you would actually be able unfold the loop. So every winded loop would actually be equal to the trivial loop, thus giving you the trivial group?
Thanks in advance.