Examples of isometries of $S^2$ Hey guys I am trying to do some practice problems for my course and I came across this problem, like I know what isometries are and how they work. But I am getting confused on how I would apply the same procedure to a sphere and I am confused on that.
1) Find two rotations of the sphere $r$, $r'$ such that $rr' = r'r$.
2) Find two rotations of the sphere $r$, $r'$ such that $rr' \neq r'r$.
3) Find an isometry of $S^2$ with no fixed points.
Please help out, itll be really appreciated thank you

1) Take $r = r'$ ;
2) take $r$ and $r'$ with orthogonal axis ;
3) $x \to - x$.
