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Chern classes and extraspecial groups. (English) Zbl 0862.20039
The determination of the mod-\(p\) cohomology algebra \(\text{H}^*(G)\) of an extraspecial \(p\)-group \(G\) is an important problem in the cohomology theory of finite groups. For \(p=2\) D. Quillen has given a complete answer [see Math. Ann. 194, 197-212 (1971; Zbl 0225.55015)]. For \(p\) an odd prime only partial results are known, they mainly concern \(\text{h}^*(G)\), the cohomology algebra \(\text{H}^*(G)\) modulo its nilradical. A subring which is somewhat more accessible than the full cohomology algebra is the Chern subring of \(\text{H}^*(G)\), and its image \(\text{ch}^*(G)\) in \(\text{h}^*(G)\). For the group \(G=p^{1+2n}_+\) the authors first exhibit Chern classes \(\alpha_1,\beta_1,\dots,\alpha_n,\beta_n\) of one dimensional representations of \(G\). Then, using Dickson invariants Chern classes \(\kappa_0,\kappa_1,\dots,\kappa_{n-1},\zeta\) are described which belong to faithful irreducible representations of \(G\) of degree \(p^n\). These two sets of elements together generate the whole of \(\text{ch}^*(G)\). The subring generated by \(\zeta,\alpha_1,\beta_1,\dots,\alpha_n,\beta_n\) had already been considered by H. Tezuka and N. Yagita [see Math. Proc. Camb. Philos. Soc. 94, 449-459 (1983; Zbl 0535.20030)]; it is known to have the same Krull dimension as \(\text{ch}^*(G)\). The authors show that \(\zeta,\kappa_0,\kappa_1,\dots,\kappa_{n-1}\) generate a different subring having the same Krull dimension as \(\text{ch}^*(G)\). The relationship between these two subrings is then explored. Formulas are given, involving Dickson invariants, that express certain powers of \(\kappa_0,\kappa_1,\dots,\kappa_{n-1}\) in terms of \(\alpha_1,\beta_1,\dots,\alpha_n,\beta_n\).

MSC:
20J06 Cohomology of groups
57R20 Characteristic classes and numbers in differential topology
20D15 Finite nilpotent groups, \(p\)-groups
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