I have two questions.


$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}\_p^r$ for some integer $r\geq 0$ and let $V=G^\*={\rm Hom}(G,{\mathbb F}\_p)$. Then $(H(G,{\mathbb F}\_p),+,\cup )$ is a ring,
$$H(G,{\mathbb F}\_p)\cong\begin{cases}S(V)&p=2\\
\Lambda (V)\otimes S(V)&p>2\end{cases}.$$
Moreover, if $p=2$ then $V={\rm Hom}(G,{\mathbb F}\_p)$ identifies as $H^1(G,{\mathbb F}\_p)$; if $p>2$ then $V$ from $\Lambda (V)$ identifies as $H^1(G,{\mathbb F}\_p)$, while $V$ from $S(V)$ identifies with the image of $V=H^1(G,{\mathbb F}\_p)$ via the Bockstein boundary map $\beta :H^1(G,{\mathbb F}\_p)\to H^2(G,{\mathbb F}\_p)$, which happens to be injective.


An alternative description is
$$H(G,{\mathbb F}\_p)\cong\begin{cases}{\mathbb F}\_2[x\_1,\ldots,x\_r]&p=2\\
\Lambda (x\_1,\ldots,x\_r)\otimes{\mathbb F}\_p[y\_1,\ldots,y\_r]&p>2\end{cases},$$
where $x\_1,\ldots,x\_r$ are a basis of $V$ and $y\_i=\beta (x\_i)$.


These results are proved via Kunneth formula.


My question is where I can find these results so that I can quote them. I saw them in a paper and in a book, but with no reference given. It seems that people regard them as "common knowledge". In the paper I mentioned the authors simply said "Recall that...", as if everybody knows this, but some need to be reminded in case they forgot.


$\bf 2.$ The second question is whether there are explicit formulas for these isomorphisms in the literature. 


If $p=2$, then the isomorphism ${\mathbb F}\_2[x\_1,\ldots,x\_r]\to H(G,{\mathbb F}\_2)$ is given by $x\_{i\_1}\cdots x\_{i\_n}\mapsto x\_{i\_1}\cup\cdots\cup x\_{i\_n}\in H^n(G,{\mathbb F}\_2)$.


If $p>2$, then the isomorphism $\Lambda (x\_1,\ldots,x\_r)\otimes{\mathbb F}\_p[y\_1,\ldots,y\_r]\to H(G,{\mathbb F}\_p)$ is given by $x\_{i\_1}\wedge\cdots\wedge x\_{i\_s}\otimes y\_{j\_1}\cdots y\_{j\_t}\mapsto x\_{i\_1}\cup\cdots\cup x\_{i\_s}\cup y\_{j\_1}\cup\cdots\cup y\_{j\_t}\in H^{s+2t}(G,{\mathbb F}\_p)$.


How about the reverse isomorphisms? Did anybody see anything published regarding this problem?


I did obtained explicit formulas for the reverse isomorphisms, where the elements of $H(G,{\mathbb F}\_p)$ are written in terms of normalized cocycles. However, I don't know wether these results are new. Somebody might have thought about them before.



Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
They're essentially exercises, compute it for $r=1$ and then invoke Künneth, and I'd expect every book to include it: Adem--Milgram's classic book for example (Corollary II.4.3 and Theorem II.4.4).


You can look at the identical homology case in Brown's classic book (Theorem V.6.6) and in particular, his description in Section V.5.3 shows how to build your reverse isomorphism (starting from a free resolution of the group module $\mathbb{F}\_pG$).


