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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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'$.


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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