A family of finite p-groups satisfying Carlson's depth conjecture

Abstract

Let p > 3 be a prime number and let r be an integer with 1 < r < p - 1. For each r, let moreover G(r) denote the unique quotient of the maximal class pro-p group of size p(r+1).We show that the mod-p cohomology ring of G(r) has depth one and that, in turn, it satisfies the equalities in Carlson's depth conjecture [2]. This is the first family of finite p-groups for which Carlson's depth conjecture has been verified besides p-groups of abelian type mod-p cohomology or extraspecial p-groups. Moreover, this computation is possible without first describing the structure of the cohomology ring.