Nilpotency degree of integral cohomology classes of \(p\)-groups (Q1865186)

Let \(p\) be a prime and \(G\) a finite \(p\)-group which is not elementary Abelian. In this interesting note it is shown that if \(\xi\in\text{Ess}(G)\), where \(\text{Ess}(G)\) is the ideal of \(H^*(G)\), the integral cohomology algebra of \(G\), consisting of the elements which restrict trivially to all proper subgroups of \(G\) then \(\xi^p=0\), provided that \(p>2\) or \(\deg(\xi)\) is even. Moreover, if \(\phi\) is a nilpotent element of \(H^*(G)\), then \(\phi^{p^{\log_p|G|-1}}=0\). A similar result in the case of mod-\(p\) coefficients was obtained by the author [in Bull. Lond. Math. Soc. 32, No. 3, 285-291 (2000; Zbl 1021.20038)].