Additive group of rationals has no minimal generating set In a comment to Arturo Magidin's answer to this question, Jack Schmidt says that the additive group of the rationals has no minimal generating set.

Why does $(\mathbb{Q},+)$ have no minimal generating set?


Let $S$ be a minimal generating set of $\mathbb{Q}$, and take $a\in S$. Let $H\leq \mathbb{Q}$ be the subgroup generated by the elements of $S$ which aren't $a$. Because $S$ is minimal, $H$ is a proper subgroup of $\mathbb{Q}$. Define $G:=\mathbb{Q}/H$. Then $G$ is non-trivial and cyclic (it is generated by the class of $a$), so $G\cong\mathbb{Z}/n\mathbb{Z}$ or $G\cong \mathbb{Z}$. This is a contradiction, because every element of $\mathbb{Q}$ is divisible by $n$, so the same should be true of a quotient of $\mathbb{Q}$.
