About direct sum of abelian groups and quotient

I'm trying to understand properly the relations between quotient and direct sum.

The first thing I wanted to know, and couldn't find online, is whether my guess is true or not: Assume $G_\alpha$ are abelian groups, and $H_\alpha \leq G_\alpha$ a subgroup, is $\frac{\bigoplus G_\alpha}{\bigoplus H_\alpha}\cong\bigoplus\frac{G_\alpha}{H_\alpha}$?

the reason i thought it would be true is because for every $G_\alpha$ we have that the only group on the nominator that affects $G_\alpha$ is $H_\alpha$. and it just seems symetric and reasonable..

thanks

Solution 1:

take the natural homomorphism $\oplus G_{\alpha}\rightarrow \oplus (G_{\alpha}/H_{\alpha})$ which is the canonical projection on each coordinate. It is clearly surjective. Now what is the kernel?