Proposition 5

A subgroup $H$ of a topological group $G$ is discrete if and only if $H$ has an isolated point. Every discrete subgroup of a Hausdorff group is closed.

If $H$ is discrete, every point of $H$ is isolated. Conversely if $H$ has an isolated point, then by translation every point of $H$ is isolated and therefore $H$ is discrete. If $H$ is discrete and $G$ is Hausdorff, then there is a neighbourhood $V$ of $e$ such that $V \cap H = \{e\}$; $\{e\}$ is closed in $G$ and therefore a fortiori in $V$, so that $H$ is locally closed at $e$. Hence $H$ is closed in $G$ by Proposition 4.