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Let $A$ be an abelian group. A set $\{ a_1 , \cdots , a_k \}$ of nonzero elements of $A$ is called linearly independent if $\sum_{i=1}^{k}n_i a_i =0$ $(n_i \in \mathbb{Z})$ implies $n_1 a_1 =\cdots =n_k a_k =0$. More explicitly, this means $n_i =0$ if $o(a_i )=\infty$ and $o(a_i )|n_i $ if $o(a_i )$ is finite.
By the rank $r(A)$ of $A$ is meant the cardinal number of a maximal independent set containing only elements of infinite and prime power orders.

Is there a finite rank torsion-free abelian group $A$ which is not finitely generated?

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Hint

The rank of the additive group of rationals is $1$.

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