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?
Which finitely generated abelian groups have balanced presentations?
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group-theory
abelian-groups
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2Cyclic 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 $C$. – 2017-01-25