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Transactions of the American Mathematical Society

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Cohomology of uniformly powerful $p$-groups
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by William Browder and Jonathan Pakianathan PDF
Trans. Amer. Math. Soc. 352 (2000), 2659-2688 Request permission

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
  • Received by editor(s): January 16, 1998
  • Published electronically: July 20, 1999
  • © Copyright 2000 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 2659-2688
  • MSC (1991): Primary 20J06, 17B50; Secondary 17B56
  • DOI: https://doi.org/10.1090/S0002-9947-99-02470-8
  • MathSciNet review: 1661313