Generators of the image of a homomorphism $G \rightarrow \operatorname{GL}_{n}(\mathbb{C})$

Question

Let $G = \langle x_1, ... , x_n \rangle$ be a finitely generated group. Given $\rho : G \rightarrow \operatorname{GL}_{n}(\mathbb{C})$, a homomorphism, we know that the image $H =\operatorname{im}\rho$ is a subgroup of $\operatorname{GL}_{n}(\mathbb{C})$.

Is $H$ then a finitely generated group? Is there a way to find its generators given we know the generators of $G$?

Answer 1

I thnik that it is obvious that $H=\operatorname{im} \rho=\langle \rho(x_1),...,\rho(x_n)\rangle$ is finitely generated with $\{\rho(x_1),...,\rho(x_n)\}$ being a set of generators

Answer 2

A finite set of generators for $H$ is $\{\rho(x_1),\ldots, \rho(x_n)\}$.

In general, a group $G$ is finitely generated if and only if there exists a surjection from a free group of finite rank $a: F_n \twoheadrightarrow G$ (this can be taken as a definition). Then if $G$ is thus finitely generated and there is a surjection $b: G \twoheadrightarrow H$, we have a surjection $ba: F_n \twoheadrightarrow H$, so that $H$ is finitely generated.