Q:

Induced homomorphism by passing to the quotient

Let $G_1$ and $G_2$ be two abelian groups with respective subgroups $N_1$ and $N_2$. and let 
$f:G_1 \to G_2$ be a homomorphism. Under what conditions is 
the induced map $f':G_1/N_1 \to G_2/N_2$ well defined?
I think the only thing to verify is that given any $g_1\in N_1$ we must have
$f(g_1)\in N_2$. is there any other condition?

A:

No other conditions. This works not just for abelian groups, but for any groups $G_1$ and $G_2$. Also, $N_2$ is not very important here. One way of thinking about this in general is: a map from $G_1$ factors through $G_1/N_1$ if and only if the normal subgroup $N_1$ is in its kernel. Now given $f:G_1\rightarrow G_2$ and a normal subgroup $N_2$ of $G_2$, you can always compose with the quotient map to get
$$
f:G_1\rightarrow G_2\rightarrow G_2/N_2,
$$
and $N_1$ is in its kernel if and only if $f(N_1)\leq N_2$.

