finite dimensional $H$–module $M$ there exists a finite dimensional rational $G$–module $N$ and a character $\chi: H \to \Bbbk$ such that:

1.  The character $\chi$ is extendible;

2.  $M_\chi \subseteq N|_H$, where $N|_H$ denotes that we consider the action of $N$ restricted to $H$.

Moreover, in the case that $M$ is a simple $H$–module, $N$ can be taken to be a simple $G$–module and even more particulary a simple $G$–submodule of $\Bbbk[G]$. Also given a pair $0 \neq m \in M$ and $z \in G$ there is such an injection $M_\chi \subseteq \Bbbk[G]$ such that $m(z) \neq 0$.

[@kn:nosotros2 Theorem 11.2.9]\[theo:obse2\] In the situation above, if the character $\chi^{-1}$ is extendible then the finite dimensional $H$ module $M$ can be imbedded (as a $H$ submodule) in a finite dimensional $G$–module $N$. In particular if for every extendible character $\chi$, the character $\chi^{-1}$ is also extendible the subgrup $H$ is observable in $G$. Moreover if $H \subseteq G$ is observable, then all characters of $H$ are extendible to $G$.

Imbed first $M_\chi \subseteq N|_H$ and then consider the inclusion of $H$–modules $\Bbbk\chi^{-1} \to
  \Bbbk[G]$ that sends $\chi^{-1} \mapsto f$ where $f$ is the polynomial guaranteeing the extendibility of $\chi^{-1}$. Clearly the 