elementary row operations

Question

We know that the elementary row operations generate the general linear group. Suppose that we have a subset of elements of a given general linear group. Is it possible to generate given general linear group via a finite set of elementary row operations?

Answer 1

It depends upon the field over which the group is defined. If you look at general linear groups $\operatorname{GL}(n,q)$ over a field of finite order $q$, then the group is itself finite, so there are only finitely many elementary row operations to begin with.

But consider the general linear group $G = \operatorname{GL}(n,\mathbb{Q})$ over the rationals $\mathbb{Q}$. Then $G$ has the multiplicative group $\mathbb{Q}^{\ast}$ of non-zero rationals as a homomorphic image (via the determinant), and $\mathbb{Q}^{\ast}$ is not finitely generated, so $G$ cannot be finitely generated.