A characterisation of the circle group

We show that if $G$ is a compact connected Abelian group such that, for some $n\in\mathbb{N}$ and some closed subgroup $H$ of $G_{(n)} = \{a\in G\mid na = 0\}$, the set $G \setminus H$ is disconnected, then $G$ is topologically isomorphic with the circle group $\mathbb{T}$.