Lemma 7

a) Let S be a subset of G such that every element of G can be written uniquely as $hs$ with $h\in H$ and $s\in S$. Then the mapping that sends $f$ to $(f(s))_{s\in S}$ is an isomorphism from the K-algebra Coind$^G_H(E)$ to the K-algebra $\mathscr{F}(S,E)$ of mappings from S to E.

b) The algebra Coind$^G_H(E)$ has finite degree over K if and only if E has finite degree over K and the index of H in G is finite. In this case, we have the formula

[Coind$^G_H(E) : K] = (G : H)[E : K]$.

c) Let $E^H$ be the algebra of invariants of the group H in E and Coind$^G_H(E)^G$ that of the group G in Coind$^G_H(E)$. The mapping $f\mapsto f(e)$ from Coind$^G_H(E)$ to E restricts to an algebra isomorphism from Coind$^G_H(E)^G$ to $E^H$.

Assertion a) follows from the definitions and implies b). By (15), a mapping from G to E is an element of Coind$^G_H(E)$ invariant under G if and only if it is constant with value an element of E invariant under H.