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A note on the homology of free abelianized extensions


Authors: Brian Hartley and Ralph Stöhr
Journal: Proc. Amer. Math. Soc. 113 (1991), 923-932
MSC: Primary 20J05; Secondary 20E22
DOI: https://doi.org/10.1090/S0002-9939-1991-1079699-X
MathSciNet review: 1079699
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Abstract: Let $ G$ be a group given by a free presentation $ G = F/N$, and $ N'$ the commutator subgroup of $ N$. The quotient $ F/N'$ is called a free abelianized extension of $ G$. We study the integral homology of $ F/N'$. In particular, if $ G$ has no elements of order $ p$ ($ p$ an odd prime), we determine the $ p$-torsion in dimension $ {p^2}$ in terms of the modulo $ p$ homology of $ G$. This extends results of Kuz'min [5, 6] describing the $ p$-torsion in smaller dimensions. Our approach is based on examining the homology of $ G$ with coefficients in symmetric powers of the augmentation ideal, which we relate to the integral homology of $ F/N'$.


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

  • [1] N. Bourbaki, Algèbre, Chapitre X: Algèbre homologique, Springer-Verlag, Paris, New York, Barcelona, and Milan, 1980.
  • [2] K. S. Brown, Cohomology of groups, Springer-Verlag, New York, Heidelberg, and Berlin, 1982. MR 672956 (83k:20002)
  • [3] Torsten Hannebauer and Ralph Stöhr, Homology of groups with coefficients in free metabelian Lie powers and exterior powers of relation modules and applications to group theory, Proc. Second Internat. Group Theory Conf. Bressanone/Brixen 1989, Rend. Circ Mat. Palermo (2) Suppl. 23 (1990), 77-113. MR 1068353 (91g:20078)
  • [4] B. Hartley and R. Stöhr, Homology of higher relation modules and torsion in free central extensions of groups, Proc. London Math. Soc. (3) 62 (1991), 325-352. MR 1085644 (92d:20074)
  • [5] Yu. V. Kuz'min, Homology theory of free abelianized extensions, Comm. Algebra 16 (1988), 2447-2533. MR 955323 (89k:20074)
  • [6] -, On some properties of free abelianized extensions, Mat. Sb. (N.S.) 180 (1989), 850-862. (Russian) MR 1015044 (90g:20077)
  • [7] R. Stöhr, On elements of order four in certain free central extensions of groups, Math. Proc. Cambridge Philos. Soc. 106 (1989), 13-28. MR 994077 (90c:20038)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1079699-X
Keywords: Homology of groups, free abelianized extensions, symmetric powers of modules
Article copyright: © Copyright 1991 American Mathematical Society

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