Topological Group is T1 iff T2

Theorem
Let $G$ be a topological group.

Then $G$ is a $T_1$ space $G$ is Hausdorff.

Necessary Condition
Follows directly from T2 Space is T1 Space.

Sufficient Condition
Let $G$ be a $T_1$ space.

Then $\left\{ {e}\right\}$ is closed.

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