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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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##### References:
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