Abstract.
Let p be an odd prime number and let n be an arbitrary positive integer. We prove that there exists a p-group 
whose mod-p cohomology ring has a nilpotent element ξ ∈ H2() satisfying ξn≠0,ξn+p−1=0. As a corollary, we exhibit a p-group whose mod-p cohomology ring contains an element of nilpotency degree n+1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Avrunin, G.S., Carlson, J.F.: Nilpotency degree of cohomology rings in characteristic two. Proc. Amer. Math. Soc. 118, 339–443 (1993)
Blackburn, N.: On a special class of p-groups. Acta Math. 100, 45–92 (1958)
Evens, L.: The cohomology of groups. Oxford Univ. Press, Oxford, 1991
Inoue, K., Kono, A.: Nilpotency of a kernel of the Quillen map. J. Math. Kyoto Univ. 33, 1047–1055 (1993)
Minh, P.A.: Essential mod-p cohomology classes of p-groups: an upper bound for the nilpotency degree. Bull. London Math. Soc. 32, 285–291 (2000)
Minh, P.A.: Evens norm and restriction maps in mod-p cohomology of p-groups. Math. Proc. Camb. Phil. Soc. 129, 253–262 (2000)
Minh, P.A.: Nilpotency degree of cohomology rings in characteristic 3. Proc. Am. Math. Soc. 130, 307–310 (2002)
Minh, P.A.: Nilpotency degree of cohomology rings in characteristic p. Proc. Am. Math. Soc. 131, 363–368 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematical Subject Classification (2000): 20J06, 20D15, 55R40
To Phuong and Nin
Rights and permissions
About this article
Cite this article
Minh, P. Any nilpotence degree occurs in mod-p cohomology rings of p-groups. Math. Z. 249, 387–400 (2005). https://doi.org/10.1007/s00209-004-0703-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-004-0703-7