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This is an exercise from J.J.Rotman's book:

Prove that the following groups are all isomorphic:

$G_1=\frac{\mathbb R}{\mathbb Z},G_2=\prod_p{\mathbb Z(p^{\infty})}, G_3=\mathbb R\oplus\big(\frac{\mathbb Q}{\mathbb Z}\big)$

What I have done is:

Since $tG_1=\frac{\mathbb Q}{\mathbb Z}$, which $t$ means the torsion subgroup; and the fact that $G_1\cong tG_1\oplus\frac{G_1}{tG_1}$ so I should show that $\frac{\mathbb R}{\mathbb Q}\cong\mathbb R$. A theorem tells me that $\frac{\mathbb R}{\mathbb Q}$ is a vector space over $\mathbb Q$ because it is abelian divisible torsion-free. The same is true for $\mathbb R$. I didn't work on a basis for any of these infinite structures good, so I can't go ahead well. :(

For $G_2$ the only first idea to me is $\frac{\mathbb Q}{\mathbb Z}\cong\sum_p{\mathbb Z(p^{\infty})}\leq G_2$.

Any helps or suggestions? Thanks.

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    @DerekHolt: Thanks so much Prof. About my first approach, I have been thinking yet: Can we easily conclude that if $\mathcal{B}$ be a basis for vector space $\mathbb R$ over $\mathbb Q$ **then** $\mathcal{B}+\mathbb Q$ is a basis for vector space $\frac{\mathbb R}{\mathbb Q}$ over $\mathbb Q$? Thanks.2012-11-13

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