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I have tried to prove the following, but to no avail...

Let $A$ be finite abelian group. For every prime $p$, prove that:

  1. $\exists r_p\ge 0:\; |A/pA|=p^{r_p}$, thus $\log_p |A/pA|$ is non-negative integer.

  2. For all primes, excluding finite number, all $r_p$ are equal.

I don't know how to tackle it. I know that $|A/pA|$ is also abelian, but don't know how to argue something about its size...

Any help will be appreciated, thanks!

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    For the first one, note that $A/pA$ is an abelian group where all elements have order $p$ (or $1$). There are two ways to go from there: Use the classification of finite abelian groups if you know it, or note that this makes the group a vector space over the field with $p$ elements. – 2017-01-21
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    For the second one: Note what happens when $p$ does not divide the order of the group. – 2017-01-21

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