Show that a group $G$ is finitely generated if and only if it is a quotient of a free group on a finite set of letters

Question

As stated in the title, I am working on this exercise:

Show that a group $G$ is finitely generated if and only if it is a quotient of a free group on a finite set of letters.

I have showed $(\Rightarrow)$ as follows:

$(\Rightarrow).$ Suppose $G$ is finitely generated, and let $S=\{a_{\alpha}\}$ be the set of all the generators. Then consider the free group on the same set of generators, denoted by $Fr(S)$. Then there exists a homomorphism $h:Fr(S)\longrightarrow G$ such that $h(a_{\alpha})=a_{\alpha}$ for all $\alpha$, and thus $h$ is surjective.

However, I don't know how to show the converse. Suppose $F$ is a free group on a finite set of letters $B$, and suppose $G\cong F/N$ for some $N$, then I know that this gives us an isomorphism $$\phi:F/N\longrightarrow G,$$ but what should I do to get the information of the generators of $G$?

Thank you!

Answer 1

The generators of $G$ are simply the images of the generators of the free group. You can see this because the homomorphism $F\to G$ is surjective; if you can get any element in the domain, the images can get any element in the codomain. In general, a quotient of a finitely generated group is finitely generated.

Notice that this need not be a minimal generating set.