Which abelian groups can be Schur Multiplier's of finitely generated groups?
Schur Multipliers of finitely generated groups
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group-theory
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1It's likely that all countable abelian groups do. But I don't know if it's done anywhere (even if it's known that every countable abelian group embeds in the center of a f.g. group).$A$first step would be to do the case of the direct sum of countably copies of $\mathbf{Q}\times\mathbf{Q}/\mathbf{Z}$, because the latter contains all countable abelian groups. If it's the center of a (perfect?) f.g. group $G$ with trivial Schur multiplier and $A$ is any central subgroup, maybe the double of $G$ over $A$, whose center is $A$, has a trivial Schur multiplier, I don't know if it's too optimistic. – 2012-10-07