Let $G$ be a finitely generated abelian group. Then prove that it is not isomorphic to $\frac{G}{N}$, for every subgroup $N\neq\langle 1\rangle$.
prove: A finitely generated abelian group can not be isomorphic to a proper quotient group of itself.
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group-theory
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1Off the top of my head, I'd look at the orders of the generators. It would seem one of them would have to drop when you move to the quotient group. – 2012-04-11