I am not sure about the connection between order of a base of free abelian group, to the order of the minimal generator set which generates the group.
I would like to inherit the conclusion from linear algabra, says that a base holds the minimal order of vectors, in order to generate a space. However, I can't find a good reason to demand the same fact about free abelian groups.
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Ron Abramovich
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1A basis of a free abelian group is also a minimal generating set (the arguing is the same as in the case of vector spaces: If it weren't minimal the elements wouldn't be linearly independent). However the converse does not hold, e.g. $(2,3)$ is a minimal generating set for $\mathbb{Z}$ but not a basis. So different minimal generating sets may have different orders. But the order will always be at least the rank (i.e. the order of a basis) of the group. – leoli1 Dec 31 '20 at 14:46