No small subgroups

Remarks

Example: $G = (\R, +)$ with $U = (-1, 1)$. If $H \subseteq U$ is a subgroup and $x \in H$ with $x \ne 0$, then $nx \in H$ for every integer $n$, and $\abs{nx} \ge 1$ for $n \ge 1/\abs{x}$, so $nx \notin U$. Hence $H = \set{0}$. The same argument with the exponential map shows that every Lie group has no small subgroups.

Non-example: the group $\Z_p$ of p-adic integers. Every neighbourhood of $0$ contains a ball $p^n \Z_p$ for some $n$, and $p^n \Z_p$ is a subgroup different from $\set{0}$.

The property is often abbreviated NSS. By the Gleason–Yamabe theorem, a locally compact group has no small subgroups if and only if it is a Lie group.