TITLE: Splitting in Short exact sequence
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: 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).