Arbitrary nilpotency degree of cohomology rings in characteristic p
Pham Anh Minh

Let $p$ be an odd prime number and let $n$ be an arbitrary positive integer. 
We prove that there exists a $p$-group $\mathcal G$ whose mod-$p$ cohomology
ring has a nilpotent element $\xi\in H^2(\mathcal G)$ satisfying $\xi^n \ne 0$,
$\xi^{n+p-1}=0$. As a corollary, we get the existence of a $p$-group whose 
mod-$p$ cohomology ring has an element of nilpotency degree $n$.