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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cohomology of uniformly powerful $p$-groups


Authors: William Browder and Jonathan Pakianathan
Journal: Trans. Amer. Math. Soc. 352 (2000), 2659-2688
MSC (1991): Primary 20J06, 17B50; Secondary 17B56
Published electronically: July 20, 1999
MathSciNet review: 1661313
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Abstract: In this paper we will study the cohomology of a family of $p$-groups associated to $\mathbb{F}_p$-Lie algebras. More precisely, we study a category $\mathbf{BGrp}$ of $p$-groups which will be equivalent to the category of $\mathbb{F}_p$-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group $G$ in this category, its $\mathbb{F}_p$-cohomology is that of an elementary abelian $p$-group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of $H^*(G ;\mathbb{Z})$ in the case that $G$ is associated to a Lie algebra $\mathfrak{L}$. To do this, we use the Bockstein spectral sequence and derive a formula that gives $B_2^*$ in terms of the Lie algebra cohomologies of $\mathfrak{L}$. We then expand some of these results to a wider category of $p$-groups. In particular, we calculate the cohomology of the $p$-groups $\Gamma _{n,k}$ which are defined to be the kernel of the mod $p$ reduction $ GL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \overset{mod}{\longrightarrow} GL_n(\mathbb{F}_p). $


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Additional Information

William Browder
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544-0001

Jonathan Pakianathan
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02470-8
PII: S 0002-9947(99)02470-8
Received by editor(s): January 16, 1998
Published electronically: July 20, 1999
Article copyright: © Copyright 2000 American Mathematical Society