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.