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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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


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:
  • MathSciNet review: 1661313