True/false: If $N\unlhd G$, then for any homomorphism $f:G\to G/N$, the kernel of $f$ contains $N$.

Question

I want to know whether the following statement is true. And if it is, how one can prove it.

"If $N\trianglelefteq G$, then for any homomorphism $f:G\to G/N$, the kernel of $f$ contains $N$."

Answer 1

It is not true in general.

For example take $$G := \bigoplus_{n \in \mathbb{N}} \mathbb{Z} \ \text{ and } \ N:= \langle (1,0,0,....) \rangle $$ Then $\varphi: G \to G/N, (a_i)_{i \in \mathbb{N}} \mapsto \overline{(0,a_1,a_2,...)}$ is an isomorphism, i.e. $\ker \varphi = 0 \nsupseteq N$.