Splitting in Short exact sequence

Question

I am trying to find whether $\{1\}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{R}\longrightarrow\mathbb{R}/\mathbb{Z}\longrightarrow \{1\}$ splits. My conjecture is it is not as we cannot find a non-zero group homomorphism from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{R}$. If my conjecture is correct, can we tweak the above sequence so that it splits?

Answer 1

Is not split:

If it is we get $\Bbb R=\Bbb R/\Bbb Z\oplus \Bbb Z$ which is not possible, because there is no element of finite order in $\Bbb R$.

Answer 2

The sequence $0 \to \mathbb{Q}\to \mathbb{R} \to \mathbb{R}/\mathbb{Q}\to 0$ splits.

To have a split one you need the subgroup to be divisible ( a $\mathbb{Q}$ vector subspace).