Group is Hausdorff iff has Closed Discrete Subgroup

Theorem
A topological group is Hausdorff it has a closed discrete subgroup.

Necessary Condition
Follows directly from Group is Hausdorff iff Identity is Closed

Sufficient Condition
Let $G$ be a Hausdorff topological group.

Let $H \leq G$ be a closed discrete subgroup.

Let $e$ be the identity of $G$.

By Set in Discrete Topology is Clopen, $\left\{{e}\right\}$ is closed in $H$.

By Closed Set in Topological Subspace: Corollary, $\left\{{e}\right\}$ is closed in $G$.

By Group is Hausdorff iff Identity is Closed, $G$ is Hausdorff.

Also see

 * Group is Hausdorff iff Discrete Subgroups are Closed