Cohomology of uniformly powerful $p$-groups

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 $GL_n(\mathbb {Z}/p^{k+1}\mathbb {Z}) \overset {mod}{\longrightarrow } GL_n(\mathbb {F}_p).$