Essential cohomology and extraspecial \(p\)-groups (Q2701688)

Let \(G\) be a \(p\)-group and \(H^*(G)\) its cohomology ring with coefficients in \(\mathbb{F}_p\). A class in \(H^*(G)\) is called essential if it vanishes upon restriction to every proper subgroup of \(G\). When computing \(H^*(G)\), the essential cohomology is usually the part that is hardest to deal with. This article considers the existence of essential cohomology for extraspecial \(p\)-groups. The case \(p=2\) was settled by \textit{D. Quillen} [Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)].NEWLINENEWLINENEWLINEThe author proves that for \(G\) an extraspecial \(p\)-group, \(p>2\), there exists non-zero essential cohomology except if \(G\) is of order~27 and exponent~3 (there is precisely one such group). As a side result, it turns out that this group is also the only one for which \(H^*(G)\) is Cohen-Macaulay. For the other groups, the author pins down explicit non-zero elements in the essential cohomology.