Cohomology of p-groups of nil potency class smaller than p

TL;DR

This paper proves that for finite p-groups with a bounded number of generators and nilpotency class less than p, the variety of their mod p cohomology algebras is finitely bounded by p and d.

Contribution

It establishes a bound on the number of isomorphism types of mod p cohomology algebras for such p-groups, linking algebraic properties to group parameters.

Findings

Number of possible cohomology algebra types is bounded by p and d.

Cohomology algebra types are finite for given p and d.

Provides a classification constraint for p-groups of small nilpotency class.

Abstract

Let $p$ be a prime number, let $d$ be an integer and let $G$ be a $d$-generated finite $p$-group of nilpotency class smaller than $p$. Then the number of possible isomorphism types for the mod $p$ cohomology algebra $H^*(G;{\mathbb F}_p)$ is bounded in terms of $p$ and $d$.