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Does there exist two abelian groups $A,B$ with an epimorphism $f: A\to B$, and two other abelian groups $A', B'$ along with an epimorphism $g: A'\to B'$ such that $A\cong A'$, $B\cong B'$ and $ker\,f \not\cong ker\,g$? It seems to me that the groups must be infinite, since we have $B\cong A/ker\,f$ and $B'\cong A'/ker\,g$.

Thanks!

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up vote 1 down vote accepted

Let me reformulate the question: you want an abelian group $G$ with two subgroups $H, H'$ which are not isomorphic but such that the quotients $G/H, G/H'$ are isomorphic.

The smallest example is $G = C_2 \times C_4, H = C_2 \times C_2, H' = 1 \times C_4$.

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