A finite presentation for a group is called balanced if is has the same number of generators as relations. Clearly cyclic groups admit balanced presentations, and $\mathbb{Z}^3$ also does. Which other finitely generated alelian groups admit balanced presentations?

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    $\begingroup$ Cyclic groups are the only finite abelian groups that admit balanced presentations, since noncyclic examples have nontrivial Schur Multipliers. I think the answer is that the examples are all of the form ${\mathbb Z}^k$ for $k \le 3$ or $C$ or ${\mathbb Z} \times C$ for a finite cyclic group $C$. $\endgroup$ – Derek Holt Jan 25 '17 at 1:40

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