Homogeneous systems of parameters in cohomology algebras of finite groups. (Q1881515)

Let \(G\) be a finite group and \(H^*(G,k)\) be the cohomology algebra of \(G\) with coefficients in a field \(k\) of characteristic \(p>0\). \textit{J. F. Carlson} [in Math. Z. 218, No. 3, 461-468 (1995; Zbl 0837.20062)] showed that the cohomology algebra \(H^*(G,k)\) has a homogeneous system \(\{\zeta_1,\dots,\zeta_r\}\) of parameters, where \(r\) is the \(p\)-rank of the group \(G\), with the property that for every \(i\) the element \(\zeta_i\) is a sum of transfers from centralizers of elementary Abelian \(p\)-subgroups of rank \(i\). The authors here construct a homogeneous system \(\{\zeta_1,\dots,\zeta_r\}\) of parameters with the additional property that for each \(i\) the system \(\{\zeta_1,\dots,\zeta_i\}\) restricts to a system of parameters in every elementary Abelian \(p\)-subgroup of rank \(i\). Moreover they show that if the \(p\)-rank of \(G\) is at most three, then the cohomology algebra \(H^*(G,k)\) has a quasi-regular sequence that is a system of parameters in \(H^*(G,k)\). The notion of a quasi-regular sequence was given by \textit{J. F. Carlson} [in Representations of algebras and related topics, Tsukuba 1990, Lond. Math. Soc. Lect. Note Ser. 168, 80-126 (1992; Zbl 0832.20076)] and it is of interest in computations of cohomology; if \(\{\zeta_1,\dots,\zeta_r\}\), where \(\zeta_i\) has degree \(n_i\), is a quasi-regular sequence forming a system of parameters in \(H^*(G,k)\), then all of the generators of \(H^*(G,k)\) lie in degrees at most \(\sum_in_i\). The authors also give a bound, in terms of the group structure of \(G\), for the number \(n\) such that all of the generators of \(H^*(G,k)\) lie in degrees at most \(n\).