Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is isomorphic to $(\mathbb{R}^n\times \Delta)/\Gamma$ where $\Delta$ is a zero dimensional compact abelian group and $\Gamma$ is a countable discrete subgroup.

The locally connected component of $G$ is the image of $\mathbb{R}^n\times\{0\}$ under this quotient and it is dense in the connected component of $G$. If the locally connected component and the connected component of $G$ coincide then they become a torus in which case $G$ can be written as $G_0\times G/G_0$.

But, clearly this does not have to be the situation.

My question is what about the case where $\Delta$ is a direct product of finite $p$-groups (not always the same prime, say $\Delta = \prod_{p\in \mathbb{Prime}} \mathbb{Z}/p\mathbb{Z}$)? Note that in this case any closed subgroup $H\leq \Delta$ satisfies that $\Delta = H\times \Delta/H$ so this might lead to the desired result.