I am searching for a reference for the proof of the following theorem.
Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = 0$ for all subgroups $H$ of $G$ and that $\hat{H}^2(H,C)$ is cyclic of order $(H : 1)$ with generator the restriction of $u$. Then the cup product with u defines an isomorphism $$x \mapsto x∪u: \hat{H}^r(G,Z) → \hat{H}^{r+2}(G,C)$$ for all $r ∈ \mathbb{Z}$.
I have found this theorem originally in the book Class field theory by Artin and Tate and the proof in there is there but seem to heavily depend on another theorem supposedly in the book (theorem 12 in chapter V is how it is worded) but from what I can tell it's not actually not in the book.
It would be great if you could either provide a proof or give me a place where I could find it. Thanks,
One way to find this result is to search for "Tate Nakayama theorem on cohomology", since it is usually attributed to both of them.
I googled this, and found Milne's book, which gives a citation (on page 3) to Serre's Corps Locaux, IX.8 . I remembered having read the proof myself somewhere, and I think this is where. (And of course, you can look in the English version Local fields if you don't like to read French.)