I am trying to find all connected subgroups of $SO(3)$ and, while solving, came up with a following question.
Let some one-dimensional subgroup $G$ of $SO(3)$ act on $S^2$ and induce a vector field by taking a derivative at $e$. By definition this vector field is smooth, so, by hairy ball theorem, there is point $p$ such that vector field at this point is zero.
Is it true that stabilizer of $p$ is one-dimensional Lie subgroup of $G$? If yes, how can one demonstrate it.
Thanks!