Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A note on Serre's theorem in group cohomology

Author: Ergün Yalçin
Journal: Proc. Amer. Math. Soc. 136 (2008), 2655-2663
MSC (2000): Primary 20J06
Published electronically: April 2, 2008
MathSciNet review: 2399026
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1987, Serre proved that if $ G$ is a $ p$-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod $ p$ cohomology algebra of $ G$, provided that the product includes at least one nontrivial class from each line in $ H^1 (G, \mathbb{F}_p)$. For $ p=2$, this gives that $ (\sigma _G )^2 =0$, where $ \sigma_G$ is the product of all nontrivial one dimensional classes in $ H^1 (G, \mathbb{F}_2)$. In this note, we prove that if $ G$ is a nonabelian $ 2$-group, then $ \sigma_G$ is also zero.

References [Enhancements On Off] (What's this?)

  • 1. F. Altunbulak, Relative extension classes and essential cohomology, Ph.D. thesis, in preparation.
  • 2. D. Benson and J. F. Carlson, The cohomology of extraspecial groups, Bull. London Math. Soc. 24 (1992), 209-235. MR 1157256 (93b:20087)
  • 3. A. Güçlükan and E. Yalçın, The Euler class of a subset complex, preprint, 2007.
  • 4. P. A. Minh, Essential mod-$ p$ cohomology classes of $ p$-groups: An upper bound for nilpotency degrees, Bull. London Math. Soc. 32 (2000), 285-291. MR 1750170 (2001c:20106)
  • 5. V. Reiner and P. Webb, The combinatorics of the bar resolution in group cohomology, J. Pure Appl. Algebra 190 (2004), 291-327. MR 2043333 (2005i:57001)
  • 6. J. P. Serre, Sur la dimension cohomologique des groupes profinis, Topology 3 (1965), 413-420. MR 0180619 (31:4853)
  • 7. J. P. Serre, Une relation dans la cohomologie des $ p$-groupes, C.R. Acad. Sci. Paris 304 (1987), 587-590. MR 897618 (88c:20064)
  • 8. Y. A. Turygin, A Borsuk-Ulam theorem for $ (\mathbb{Z} \sb p)\sp k$-actions on products of $ (\operatorname{mod}p)$ homology spheres, Topology Appl. 154 (2007), 455-461. MR 2278695 (2007j:55005)
  • 9. E. Yalçın, Group actions and group extensions, Trans. Amer. Math. Soc. 352 (2000), 2689-2700. MR 1661282 (2000j:57076)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 20J06

Retrieve articles in all journals with MSC (2000): 20J06

Additional Information

Ergün Yalçin
Affiliation: Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey

Keywords: Cohomology of groups, Stiefel-Whitney classes, essential cohomology
Received by editor(s): March 12, 2007
Published electronically: April 2, 2008
Additional Notes: The author was partially supported by TÜBİTAK-BDP and TÜBA-GEBİP/2005-16.
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society