Let $F$ be the free abelian group generated by $X$, where $X=\{a,b_{1},b_{2},...,b_{n}...\}$ and let $K=\langle Y \rangle$. Here $Y=\{2a, a-2^{n}b_{n},n\ge 1\}$. Define $G=F/K$. Now I am supposed to prove:
$0\rightarrow \langle a \rangle \rightarrow G\rightarrow \bigoplus_{n\ge 1}I_{2^{n}}\rightarrow 0$
I am not sure how to construct this map. For example, with $2a\approx 0$ we should have the inclusion map to be $na\rightarrow n\pmod{2}a$. But this cannot be injective.
A better choice is to map $a=2^{n}b_{n}$ to individual coordinates, since $2a=1$ we effectively map $a$ to $2^{n-1}$ in all $I_{2^{n}}$ coordinates. Now let the second map be the projection map that projects arbitrary element $g\in G$ to individual coordinates. This map is clearly surjective. But if I choose as above, then the sequence will not be exact. What is a good choice?
Then I am suppose to prove $Hom(\mathbb{Q},G)=0$ and I do not know how to prove it.