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
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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.

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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.