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

DOI:
https://doi.org/10.1090/S0002-9947-99-02470-8

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

Received by editor(s):
January 16, 1998

Published electronically:
July 20, 1999

Article copyright:
© Copyright 2000
American Mathematical Society