Summary
The mod-p cohomology ring of the extraspecialp-group of exponentp is studied for oddp. We investigate the subquotient ch(G) generated by Chern classes modulo the nilradical. The subring of ch(G) generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them.
Similar content being viewed by others
References
M.F. Atiyah. Characters and cohomology of finite groups.Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64
D. J. Benson.Polynomial Invariants of Finite Groups. London Math. Soc. Lecture Note Ser. no. 190 (Cambridge Univ. Press, 1993)
D. J. Benson and J. F. Carlson. The cohomology of extraspecial groups.Bull. London Math. Soc. 24 (1992), 209–235. Erratum:Bull. London Math. Soc. 25 (1993), 498
D. J. Green. Chern classes and extraspecial groups of orderp 5. Preprint, 1995
I. J. Leary. The mod-p cohomology rings of somep-groups.Math. Proc. Cambridge Philos. Soc. 112 (1992), 63–75
I. J. Leary and N. Yagita. Some examples in the integral and Brown-Peterson cohomology ofp-groups.Bull. London Math. Soc. 24 (1992), 165–168
G. Lewis. The integral cohomology rings of groups of orderp 3.Trans. Amer. Math. Soc. 132 (1968), 501–529
D. Quillen. The spectrum of an equivariant cohomology ring: I.Ann. of Math. (2)94 (1971), 549–572
D. Quillen. The mod-2 cohomology rings of extra-special 2-groups and the spinor groups.Math. Ann. 194 (1971), 197–212
D. Quillen and B. B. Venkov. Cohomology of finite groups and elementary abelian subgroups.Topology 11 (1972), 317–318
M. Tezuka and N. Yagita. The varieties of the modp cohomology rings of extra specialp-groups for an odd primep.Math. Proc. Cambridge Philos. Soc. 94 (1983), 449–459
Author information
Authors and Affiliations
Additional information
The support of the Deutsche Forschungsgemeinschaft and the Max Kade Foundation is gratefully acknowledged, as is the kind hospitality of the ETH Zürich.
The support of the Eidgenössische Technische Hochschule in Zürich and of the Leibniz Fellowship Programme is acknowledged with thanks.
This article was processed by the author using theLatex style filecljour1 from Springer-Verlag.
Rights and permissions
About this article
Cite this article
Green, D.J., Leary, I.J. Chern classes and extraspecial groups. Manuscripta Math 88, 73–84 (1995). https://doi.org/10.1007/BF02567806
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02567806