Homorphism between groups and generators

Question

Let $f: G \rightarrow H$ be a homomorphism of finite groups.

Since $G$ is finitely generated, $G = \langle x_1 , ... , x_n \rangle$. Is it then true that $H = \langle f(x_1), ... , f(x_n) \rangle$? Or is this only true when $f$ is an isomorphism?

Answer 1

Consider the homomorphism $\iota: \Bbb Z_3\to \Bbb Z_6$ given by $\iota(g)=0$ for all $g\in \Bbb Z_3$.

Hint: Consider $|\Bbb Z_3|$ and $|\Bbb Z_6|$.

Also, consider $\pi: \Bbb Z_2\times\Bbb Z_3\to \Bbb Z_3$ given by $(g,h)\mapsto h$. Then $\pi$ is surjective but your property holds.

In fact, it is true if and only if $f$ is surjective.

Answer 2

You should prove that, in fact:

$$\langle f(x_1), \dots, f(x_n)\rangle = \operatorname{im}(f).$$

This means that $H$ is generated by the $f(x_i)$ exactly when $H = \operatorname{im}(f)$, i.e. when $f$ is surjective.