A non-closed $p$-group.

Question

Are there a Hausdorff topological group $(G,\mathcal T)$ and and a non-closed $p$-group $P\le G$ ?

a $p$-group where $p$ is a prime number, is a group $P$ such that $$(\forall a\in P)(\exists n\in \Bbb N)(|a|=p^n)$$

Answer 1

Yes, let $G = S^1$ the circle group, and $P$ the subgroup of elements whose order is a power of $p$ (where $p$ is the arbitrarily chosen prime). Since

$$P = \left\{\exp \left(\frac{2\pi i k}{p^n}\right) : k \in \mathbb{N}, n \in \mathbb{N}\right\},$$

$P$ is a dense subgroup of $G$, and since $P \neq G$, it is hence not closed.