A simple question on Ext groups

Question

Let $G$ be a finite abelian group. Is $\mathrm{Ext}_{\mathbb{Z}}(G,\mathbb{R}/\mathbb{Z})$ trivial? If not, under what condition is it trivial?

Answer 1

Let $A$ be an abelian group. The following are equivalent:
1. $A$ is divisible.
2. $A$ is an injective $\mathbb{Z}$-module.
3. $\operatorname{Ext}_{\mathbb{Z}}^i(G,A) = 0$ for every abelian group $G$.

Note that $\mathbb{R}$ is a divisible group and since divisibility is preserved under taking quotients, $\mathbb{R} / \mathbb{Z}$ is also divisible.

Answer 2

Yes. $\mathbb{R}/\mathbb{Z}$ is injective.