is also $H$–fixed. It we write $f_1=f_2g$ for $g$ as above, using the fact that $f_1$ and $g$ are fixed by $H$, we conclude that $f_2$ is also fixed.

Next we prove Theorem \[theo:geocarac\].

From the following general fact (see [@kn:nosotros2 Theorem 1.4.48]): if $C \subseteq X$ is a closed subset of a quasi–affine variety, then there is a global section $0\neq f \in \mathcal O_X(X)$ such that $f|_C=0$, and the second assertion of Theorem \[theo:charcinv\], it follows directly that if $G/H$ is quasi affine then $H$ is observable in $G$.

We sketch the proof of the converse assertion and we work in the case that $G$ is irreducible (it is easy to show that is enough to treat this particular situation).

Assume that $H \subseteq G$ is observable, and using the fact that ${}^H[\Bbbk[G]]=[{}^H\Bbbk[G]]$ (see Observation \[obse:obse3\]) we can take a family of field generators of the invariant rational functions $\{f_1,\cdots,f_n\} \subseteq
    {}^H[\Bbbk[G]]=[{}^H\Bbbk[G]]$ that are of the form $f_i=u_i/u_0$ with $\{u_0,\cdots,u_n\} \subseteq {}^H\Bbbk[G]$ for $i=1,\cdots,n$. Let $N$ the finite dimensional rational $G$–module generated by $\{u_0,\cdots,u_n\}$, $M=\bigoplus_{i=0}^nN$ and take $m_0=(u_0,\cdots,u_n) \in M$. It is a standard result in the theory of affine algebraic groups that $H=\{x \in G: x\cdot f=f 