I am trying to read the article "Three-dimensional affine crystallographic groups" of Fried–Goldman (Adv. Math., 1983). At some place, it states that if $G$ is a connected solvable closed subgroup of $\operatorname{GL}(\mathbb R^n)$, then its normalizer in $\operatorname{GL}(\mathbb R^n)$ is an algebraic group. I couldn't see how this works. I would like to know if this is a classical fact, and if there is any reference on that ?

## 1 Answer

$\begingroup$
$\endgroup$

The normalizer of every connected closed subgroup of $\mathrm{GL}_n(\mathbf{R})$ is the stabilizer of its Lie algebra, so is Zariski-closed.

any(Zariski-closed) linear subgroup of a linear group is linear.) $\endgroup$2more comments