Isomorphism between $G$ and $\mathbb{Q}^{*}$ Let $\{G_{n}\}_{n\in \mathbb{N}}$ be a family of additive groups with $G_{1}=\mathbb{Z}_{2}$ and $G_{n}=\mathbb{Z}$ for $n\geq 2$
$$G=\bigoplus_{n\in \mathbb{N}}G_{n}$$
I want to prove that $G\cong \mathbb{Q}^{*}$
I can't find the correct function, can you give me a hint?
 I tried :
$$(g_{1},g_{2},...)\longrightarrow -1^{g_{1}}\sum_{n\geq 2}\frac{g_{n}}{10^{n-2}} $$

Hint: Think of the $G_n$'s for $n>1$ as being indexed by the prime numbers, and use prime factorization.
