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It can be easily shown that, the Prüfer $p$-group $\mathbb Z(p^\infty)$ is isomorphic to multiplicative group $$R_p=\{e^{2\pi ik/p^n}|k\in\mathbb Z,n\geq0\}$$ Now I want to prove that:

$$\frac{\mathbb Q}{\mathbb Z}\cong\sum_p \mathbb Z(p^\infty)$$

What I did is to consider the following map: $$\frac{\overbrace{p_1^{a_1}p_2^{a_2}...}^{k_i}}{p_i^{a_i}}+\mathbb Z\to \big(e^{2\pi ik_1/p_1^{a_1}},e^{2\pi ik_2/p_2^{a_2}},...\big)$$ Do you know another map? :-) Thanks.

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Your question is equivalent to asking for automorphisms of $\mathbb Q/\mathbb Z$. Can you give examples of these? –  Mariano Suárez-Alvarez Oct 13 '12 at 6:46
    
@MarianoSuárez-Alvarez: Honestly, I didin't think about this point of view. You say my map is not a practical map? –  Babak S. Oct 13 '12 at 7:04
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Your map looks fine: the point was that once you have one isomorphism between $G$ and $H$, asking for another is equivalent to asking for automorphisms of either $G$ or $H$, because given one of those we just compose it on to the isomorphism we already have to get a new one. Hence the question: can you think of any automorphisms of $\mathbb{Q}/\mathbb{Z}$? –  Kevin Carlson Oct 13 '12 at 7:49
    
Look up profinite completions and Pontryagin duality. –  Urban Oct 13 '12 at 8:04
    
Thanks @MarianoSuárez-Alvarez. –  Babak S. Oct 13 '12 at 9:38

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