Lie groups have no small subgroups

Choose an injective local exponential chart on a ball of radius $r$ in the Lie algebra, and let $0<\epsilon <r/2$. The neighborhood $\exp (B_{\epsilon })$ contains no nontrivial subgroup. Otherwise it contains an element $\exp X$ with $0<\Vert X\Vert <\epsilon$. Choose the first integer $m$ with $m\Vert X\Vert \ge \epsilon$. Then $\epsilon \le m\Vert X\Vert <2\epsilon <r$, so chart injectivity puts $(\exp X{)}^m=\exp (mX)$ outside $\exp (B_{\epsilon })$, a contradiction. Zero-dimensional Lie groups have the singleton identity as such an open neighborhood.