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What is the difference between a "free abelian group" and a free group that is abelian or an abelian group that is free.. it seems very confusing because in my understanding a free group $G$ is a group that has a basis $B=(b_i)_i$, that is $\forall g\in G$, $g=\prod{b_i^{m_i}}$ ,$m_i\in \mathbb Z$. In particular when $G$ is abelian, we write this additively and we have $g=\sum_{i}{m_ib_i}$, $m_i\in\mathbb Z$ and we call this a "free abelian group" is this correct?

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