deciding whether a group homomorphism preserving some properties or not as follows

Question

Let $f:G \to H$ be a group homomorphism. Which the statement is true? Why?

1. If $H$ is finite, then so is $G$.

2. If $H$ is finite cyclic, then so is $G$.

3. If $H$ is abelian, then so is $G$.

Edit: $f$ isn't trivial group homomorphism.

My attempt:

1. If $G$ is finite, then so is $H$, but not as converse. The converse is hold iff $f$ is injective.

2. Same with point 1) above.

3. Same with point 1) above and let $f(a),f(b)\in H$. Then, $f(a)f(b) = f(b)f(a) \Leftrightarrow f(ab)=f(ba)$. Note that $ab=ba$ is not necessarily true.

Any idea? Thanks in advanced.

Answer 1

Remember that there is a trivial group homomorphism between any two groups, namely the function $f\colon G\to H$ defined by $f(g) = e_H$ for all $g\in G$ (here $e_H$ denotes the identity element of $H$).

With this in mind, is there any information about the group $H$ that would give us information about the group $G$?