Prove that $\displaystyle <\mathbb Q, +> $ group has no finite set of generators.

Any help would be appreciated!

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- Feb 20th 2011, 06:02 AMdougGroup theory
Prove that $\displaystyle <\mathbb Q, +> $ group has no finite set of generators.

Any help would be appreciated! - Feb 20th 2011, 06:24 AMroninpro
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.