to $G$ and the geometric structure of the homogeneous space $G/H$.

It is substantially harder to study homogeneous spaces in the category of algebraic groups than for example in the closely related category of Lie groups. The basic general results concerning the existence of a natural structure of algebraic variety on $G/H$ are due to M. Rosenlicht and A. Weil in the mid 1950s (see [@kn:Roshomo] and [@kn:Weilhomo]). The proof that $G/H$ is quasi–projective is due to W. Chow and appeared in 1957 (see [@kn:ChowPQ]).

The proof that the quotient of an affine group by a normal closed subgroup is also an *affine* algebraic group seemed to have appeared for the first time in 1951[^3], in Chevalley’s very important foundational book, [@kn:Che].

In ERA the following theorem –that provides a very precise characterization of observability in geometric terms– is proved.

\[theo:geocarac\][@kn:hmbb Theorem 4] If $H \subseteq G$ is a closed inclusion of affine algebraic groups, then $H$ is observable in $G$ if and only if the homogeneous space $G/H$ is a quasi–affine variety.

In particular, the above theorem guarantees that a normal subgroup is always observable and hence, that the normality hypothesis unavoidable for the situation of Lie groups as 