COHOMOLOGY OF UNIFORMLY POWERFUL p-GROUPS

In this paper we will study the cohomology of a family of p-groups associated to Fp-Lie algebras. More precisely we study a category BGrp of p-groups which will be equivalent to the category of Fp-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group G in this category, its Fp-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;Z) in the case that G is associated to a Lie algebra 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 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 Γn,k which are defined to be the kernel of the mod p reduction GLn(Z/pZ) mod −→ GLn(Fp). 1991 Mathematics Subject Classification. Primary: 20J06, 17B50; Secondary: 17B56.