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

[Euge]

Here's this week's problem!

_____________

Problem. Show that the short exact sequence of abelian groups

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

is not split exact. (The sums and products are extended over all prime numbers $p$.)
_____________Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Euge]

No one answered this week's problem. Here is my solution.

Spoiler

Let

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

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.