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${\rm Hom}(\mathbb Z, \sum_{p\in P}\mathbb Z_p)$ and $\prod_{p\in P}{\rm Hom}(\mathbb Z, \mathbb Z_p)$ are not isomorphic where $P$ is the set of all primes.

I was checking the elements of each to see if they have finite order or infinite order. The direct product has elements of both finite order and infinite order. But this led me nowhere.

Any hint?

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    $\begingroup$ I seem to get it. If i am correct, the elements of the former are all of finite order as I can map the integer 1 to any element of the direct sum which has finite order. $\endgroup$
    – scsnm
    Commented Jan 6, 2022 at 12:51
  • $\begingroup$ ${\rm Hom}(\Bbb Z,\bigoplus_{p\in P}\Bbb Z_p)=\bigoplus_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$ and ${\rm Hom}(\Bbb Z,\prod_{p\in P}\Bbb Z_p)=\prod_{p\in P}{\rm Hom}(\Bbb Z,\Bbb Z_p)$ $\endgroup$
    – reuns
    Commented Jan 6, 2022 at 15:55

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