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

TL;DR

This paper investigates the mod-p cohomology rings of specific finite p-groups, demonstrating they have depth one and satisfy Carlson's conjecture, thus providing new examples supporting the conjecture.

Contribution

It constructs a family of finite p-groups with cohomology rings that satisfy Carlson's depth conjecture, expanding known cases.

Findings

Cohomology rings of G_r have depth one

G_r satisfy Carlson's depth conjecture

Provides explicit examples of p-groups with these properties

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 [3].