Let $N\unlhd G$ with $[G:N]=3$. Find all non-trivial group homomorphisms $f:G\to\Bbb{Z}/n\Bbb{Z}$ for $n=2,3,...,10$ with $N\le\ker(f)$.

Question

Let $G$ be a group and $N$ a normal subgroup of $G$ with index 3, $[G:N]=3$. I need to find all non-trivial group homomorphisms $f:G\rightarrow \mathbb{Z}/n\mathbb{Z}$ for $n=2,3,...,10$ with $N\leq \text{ker}(f)$.

How do I do this, I have no idea how to even start? I mean for example taking $n=2$, I would have $\text{ker}(f)=2\mathbb{Z}$. Now I need to find a normal-subgroup of $2\mathbb{Z}$ with $[G:N]=3$ right? How do I do that?

Answer 1

Note that any homomorphism $f\colon G\to \mathbb Z/n\mathbb Z$ with $N\le \ker(f)$ induces a homomorphism $\overline{f}\colon G/N\to\mathbb Z/n\mathbb Z$ given by $\overline f(\overline g) = f(g)$, where $\overline g = gN$. What do you know about the group structure of $G/N$ and does that help you find all homomorphisms to $\mathbb Z/n\mathbb Z$ for the given list of $n$s?