The additive group of the rationals does not have an automatic presentation

We prove that the additive group of the rationals does not have an automatic presentation. The proof also applies to certain other abelian groups, for example, torsion-free groups that are $p$-divisible for infinitely many primes $p$, or $Q/Z$. The proof is combinatorial and uses most notably Freiman's theorem on sets with small doubling.