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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the cohomology of split extensions
of finite groups


Author: Stephen F. Siegel
Journal: Trans. Amer. Math. Soc. 349 (1997), 1587-1609
MSC (1991): Primary 20J06
DOI: https://doi.org/10.1090/S0002-9947-97-01747-9
MathSciNet review: 1376556
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Abstract: Let $G=H\rtimes Q$ be a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentials $d_2$ in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a field $k$. We generalize this to give an explicit description of all the $d_r$ ($r\geq 2$) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in the $kG$-endomorphism algebra of the minimal $kH$-projective resolution induced from $H$ to $G$. This twisting cochain not only determines the differentials, but also allows one to construct an explicit $kG$-projective resolution of $k$.


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Additional Information

Stephen F. Siegel
Email: siegel@math.umass.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01747-9
Received by editor(s): October 30, 1995
Additional Notes: The author was supported by a Sloan Foundation dissertation fellowship and a National Science Foundation postdoctoral fellowship.
Article copyright: © Copyright 1997 American Mathematical Society

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