Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$.
Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime.
Denote by $G[a]$ resp. $G[b]$ the $a$-resp. $b$-torsion subgroups of $G$.
Then why are the following maps isomorphisms:
(1) $G[a] \times G[b] \rightarrow G$ (addition map)
(2) $G \rightarrow G[a]\times G[b], g\mapsto (b\cdot g,a\cdot g)$ ?
Note that I don't assume finiteness of $G$ and that injectivity of both maps are clear to me. The problem is surjectivity.