A note on Serre’s theorem in group cohomology
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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.References
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Additional Information
- Ergün Yalçın
- Affiliation: Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
- Email: yalcine@fen.bilkent.edu.tr
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2655-2663
- MSC (2000): Primary 20J06
- DOI: https://doi.org/10.1090/S0002-9939-08-09408-2
- MathSciNet review: 2399026