Why is the short exact sequence of abelian groups not split exact?

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Here's this week's problem!

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Problem. Show that the short exact sequence of abelian groups

\(\displaystyle 0 \rightarrow \bigoplus_p \Bbb Z/(p) \rightarrow \prod_p \Bbb Z/(p) \rightarrow \frac{\prod_p \Bbb Z/(p)}{\bigoplus_p \Bbb Z /(p)} \rightarrow 0\)

is not split exact. (The sums and products are extended over all prime numbers $p$.)
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No one answered this week's problem. Here is my solution.

Spoiler

Let

\(\displaystyle A = \bigoplus_p \Bbb Z/(p) \quad \text{and} \quad B = \prod_p \Bbb Z/(p).\)

It is enough to show that there is no isomorphism from $A \oplus B/A$ onto $B$. Suppose to the contrary that there is such an isomorphism, call it $f$. Then the composition $B/A \xrightarrow{\iota} A \oplus B/A \xrightarrow{f} B$ is zero, which contradicts the fact that $f$ is one-to-one.