\text{ for all } f \in {}^H[\Bbbk[G]]\}$ (see for example [@kn:nosotros2 Corollary 8.3.4]) and then in our case we have that $H=G_{m_0}$ the stabilizer of $m_0$. It can be proved that $G/H$ is isomorphic to the $G$–orbit of $m_0$ in $M$ (result that is obvious in the case of zero characteristic, but that in general a proof of the separabililty of the action in this situation is needed) and as such it is a quasi–affine variety (for more details see [@kn:nosotros2 Section 8.3]).

Observability and Hilbert’s $\boldsymbol{14}^{\text{th}}$ problem
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About ten years after the introduction of the concept of observability, an important relation with the so called Hilbert’s $14^{\text{th}}$ problem was discovered by G. Grosshans in [@kn:gross14]. As such, the concept of observability became another important element in the toolkit of invariant theory.

We describe briefly some parts of the contents of the important paper mentioned above, wherein the author distinguishes three situations –that he names as “the main problems”[^4].

[Problem 1. Galois characterization of the observable subgroups.]{}

The author presents an interesting new perspective of the concept of observable subgroup.

\[defi:viagalois\] If $G$ is an affine connected algebraic group, define the sets $\mathfrak H=\{H: H \subseteq G \text{ is 