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