Necessary and Sufficient Condition for cyclic $H$ to be a homomorphic image of cyclic $G$

Question

Let $G$ and $H$ be cyclic groups. I need to find and prove a necessary and sufficient condition in order for $H$ to be homomorphic to $G$.

By the Fundamental Theorem on Homomorphisms, the possible homomorphic images of $G$ are isomorphic to $G/N$, where $N$ is a normal subgroup of $G$.

So, is a necessary and sufficient condition for $H$ to be the homomorphic image of $G$ that $H \simeq G/N$?

I was also told to use the following theorem to describe such a necessary and sufficient condition:

Theorem on the Classification of Cyclic Groups:

1. An infinite cyclic group is isomorophic to $\mathbb{Z}$.
2. A finite cyclic group is isomorphic to $\mathbb{Z}_{n}$ for some $n \geq 1$.
3. Groups $\mathbb{Z}_{n}$ and $\mathbb{Z}_{m}$ are not isomorphic for distinct $m$ and $n$.

How does this fit together with my necessary and sufficient condition (assuming it's correct)?

If my necessary and sufficient condition is not correct, what is a correct condition? And how would one go about proving such a condition?

Thank you.

Answer 1

Well, this is correct (and even more general): $H$ is a homomorphic image of $G$ iff there is an $N\trianglelefteq G$ such that $H\cong G/N$.

However, using that a cyclic group is determined by its cardinality, we can give a straighter answer:

1. Each cyclic group is a homomorphic image of $\Bbb Z$, the infinite cyclic group.
2. If $n|m$, we have a surjective homomorphism $\Bbb Z_m\to\Bbb Z_n$.
3. If $\Bbb Z_n\cong\Bbb Z_m/N$ for any (normal) subgroup $N$ of $\Bbb Z_m$, we have $n|m$.