Prove that $\displaystyle <\mathbb Q, +> $ group has no finite set of generators.
Any help would be appreciated!
For the ease of notation, suppose there are two generators, $\displaystyle m_1/n_1, m_2/n_2$. Let $\displaystyle p$ be a prime such that $\displaystyle \gcd(n_1, n_2,p)=1$. Is it possible to write $\displaystyle 1/p$ as a combination of $\displaystyle m_1/n_1, m_2/n_2$?
You can generalize this argument for any supposed number of generators.