Hostname: page-component-7f64f4797f-bnl7t Total loading time: 0 Render date: 2025-11-04T08:18:31.182Z Has data issue: false hasContentIssue false
  • English
  • Français

Cup products and group extensions

Part of: Connections with homological algebra and category theory Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

P. A. Linnell
P. A. Linnell
Affiliation:
Department of Mathematics Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0123. U.S.A.E-mail address: LINNELL@VTMATH.BITNET
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a finitely generated group and let R be a commutative ring, regarded as a G-module with G acting trivially. We shall determine when the cup product of two elements of H1(G, R) is zero. Our method will use the interpretation of H2(G, R) as extensions of G by R. This will give an alternative demonstration of results of Hillman and Würfel.

Keywords

cup productgroup extension

MSC classification

Secondary: 20J05: Homological methods in group theory 20E22: Extensions, wreath products, and other compositions

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 1 , February 1991 , pp. 108 - 115
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Benson, D. J., Modular representation theory, (Lecture Notes in Mathematics, vol. 1081, Springer-Verlag, Berlin, New York, 1984).Google Scholar
[2]Curtis, C. W. and Reiner, I., Methods of integral representation theory, vol. 1, (Wiley-Interscience, New York, 1981).Google Scholar
[3]Hillman, J. A., ‘The kernel of the cup product’, Bull. Austral. Math. Soc. 32 (1985), 261–274.CrossRefGoogle Scholar
[4]Hillman, J. A., ‘The kernel of integral cup product’, J. Austral. Math. Soc. (Ser. A) 43 (1987), 10–15.CrossRefGoogle Scholar
[5]Hilton, P. J. and Stammbach, U., A course in homological algebra (GTM 4, Springer-Verlag, Berlin, New York, 1971).CrossRefGoogle Scholar
[6]Lane, S. Mac, Homology, (Springer-Verlag, Berlin, New York, 1975).Google Scholar
[7]Würfel, T., ‘A note on the cup product for pro-π groups’, Proc. Amer. Math. Soc. 102 (1988), 809–813.Google Scholar