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I want to show that:

If $G=\prod_{p\in P}\mathbb Z_p$, wherein $P$ is the set of all primes, then $\frac{G}{tG}$ is divisible.

I know that $tG$ is not a direct summand and if $x\in G$ wants to be divisible by every primes, it will be $0$. I wanted to know $tG$ for myself first, so I took an element in it: $f=(a_1,a_2,...)\in tG\longrightarrow\exists n, nf=0 $ Can I conclude here that for infinitely many $a_i\in \mathbb Z_{p_i}$, we necessarily have $a_i=0$? Is there any formal well-known description for $tG$? Thanks for any hint or references of where to start the main problem above.

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    @BabakSorouh, this'll work with Derek's adjustment for showing that $G/tG$ is divisible by primes-now to show that it's divisible by all $n$, you just have one more step in passing to the prime factorization of $n$. Then if you want $x\in G/tG$ such that $nx=g,$ you can find a solution in all but the (finitely many) $p_i$ which divide $n$, and set the others to 0. Feel free to answer your own question, now.2012-10-12

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