Cohomology of uniformly powerful $p$-groups

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