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Suppose $A_1$, $A_2$, $A_3$ and $B_1$, $B_2$, and $B_3$ are two short exact sequences of abelian groups.

I am looking for two such short sequences where $A_1$ and $B_1$ is isomorphic and $A_2$ and $B_2$ are isomorphic but $A_3$ and $B_3$ are not.

(Similarly I would like examples in which two of the other pairs are isomorphic but the third pair are not, etc)

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up vote 3 down vote accepted

Here are some finite examples:

$$\begin{array}{c} 0 \to& C_2\times C_2 &\to& C_2\times C_2\times C_4 &\to& C_4 &\to 0 \\ 0 \to& C_2\times C_2 &\to& C_2\times C_2 \times C_4 &\to& C_2\times C_2 &\to 0 \\ 0 \to& C_4 &\to& C_2\times C_2 \times C_4 &\to& C_2 \times C_2 &\to 0 \\ 0 \to& C_4 &\to& C_2\times C_8 &\to& C_2 \times C_2 &\to 0 \end{array}$$

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For the first pair take $$0 \longrightarrow \mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0$$ and $$0 \longrightarrow \mathbb{Z} \stackrel{3}{\longrightarrow} \mathbb{Z} \longrightarrow \mathbb{Z} / 3\mathbb{Z} \longrightarrow 0.$$

For sequences with non-isomorphic first pairs you can use an infinite direct sum of $\mathbb{Z}$'s and include one or two copies of $\mathbb{Z}$. The quotient will be the infinite direct sum again so the second and third pairs are isomorphic but the first pair will be non-isomorphic.

Finally for non isomorphic central pairs take $$0 \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 2\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0$$ and $$0 \longrightarrow \mathbb{Z} / 2\mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} / 4\mathbb{Z} \longrightarrow \mathbb{Z} / 2\mathbb{Z} \longrightarrow 0.$$

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