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 $S\subseteq\mathbb Q$ be such that $\langle S\rangle=\mathbb Q$. Fix $a\in S$, and put $T=S\setminus\{a\}$, let us see that also $\langle T\rangle=\mathbb Q$. We have $$\frac{a}{2}=a\cdot k_0+\sum_{i=1}^na_i\cdot k_i,$$
for some $k_i\in\mathbb Z$ and $a_i\in T$. Then $$a=a\cdot (2k_0)+\sum_{i=1}^na_i\cdot (2k_i),$$ that is, $$a\cdot m=\sum_{i=1}^na_i\cdot (2k_i),$$ where $m=1-2k_0$ is nonzero; as $k_0$ is an integer. 
Now $\frac{a}{m}$ can be expressed as a combination of elements of $S$, say $\frac{a}{m}=a\cdot r_0+\sum_{i=1}^lb_i\cdot r_i,$ with $b_i\in T$,$r_i\in\mathbb Z$, thus $$a=a\cdot mr_0+\sum_{i=1}^lb_i\cdot mr_i=\sum_{i=1}^na_i\cdot r_0(2k_i) +\sum_{i=1}^lb_i\cdot mr_i.$$
