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Cohomology of uniformly powerful -groups

Authors:	William Browder Jonathan Pakianathan

Institution:	Department of Mathematics, Princeton University, Princeton, New Jersey 08544-0001 Jonathan Pakianathan ; Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Abstract:	In this paper we will study the cohomology of a family of -groups associated to $\mathbb{F}_p$ -Lie algebras. More precisely, we study a category $\mathbf{BGrp}$ of -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 in this category, its $\mathbb{F}_p$ -cohomology is that of an elementary abelian -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 is associated to a Lie algebra $\mathfrak{L}$ . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of $\mathfrak{L}$ . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups $\Gamma _{n,k}$ which are defined to be the kernel of the mod reduction $GL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \overset{mod}{\longrightarrow} GL_n(\mathbb{F}_p).$

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