A group $G$ is indecomposable if $G \neq \langle e \rangle$ and $G$ cannot be written as the direct product of two of its proper subgroups. Why is the additive group of rational numbers $(\mathbb{Q},+)$ indecomposable?
[Math] Why is the additive group of rational numbers indecomposable
abstract-algebragroup-theory
Best Answer
I suspect, what you want to ask is why $(\mathbb{Q},+)$ is indecomposable, ie. it cannot be written as the direct sum of two subgroups.
The answer is that two non-trivial subgroups must intersect non-trivially. If $\{0\}\neq H, K < \mathbb{Q}$, then choose non-zero $p/q \in H, a/b\in K$, then $$ qa\frac{p}{q} = ap = pb\frac{a}{b} \in H\cap K\setminus \{0\} $$