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?


factor out the integers, and consider $Q_*$ the additive rationals modulo unity.
each prime $p$ may be associated with a 'pure' subgroup $Q_{*p}$containing all (and only) the fractions whose denominator is a power of $p$. such a subgroup is the limit by set union of an ascending chain of subgroups generated by a monotone sequence of (negative) powers of $p$, for example: 
$$
<\frac{1}{p}>\; \subset \;<\frac{1}{p^2}> \; \subset \dots
$$
any cofinal sequence will do, but any non-cofinal (i.e. finite, here) sequence must exclude denominators with a power of $p$ greater than the maximal power in the finite chain.
thus it is trivial that for a pure subgroup no generator is indispensible.
but $Q_*$ is the direct sum of its pure subgroups. so by an easy argument, no generator can be indispensible.
