Corollary

— Let $\psi$ be the K-linear mapping of $N \otimes_K N$ into the product vector space $N^\Gamma$ such that $\psi(x \otimes y)\,(a) = \sigma(x)\,y$ for $x,\,y$ in $N$ and $\sigma \in \Gamma$. Then $\psi$ is injective and it is bijective when $\Gamma$ is finite.

This is the particular case $M = N$ of Prop. 8.