# Rank, coclass and cohomology

## TL;DR

This paper proves that for any prime p, finite p-groups with fixed coclass have finitely many mod-p cohomology rings, confirming a conjecture by Carlson through a stronger bounded rank result.

## Contribution

It establishes the finiteness of mod-p cohomology rings for p-groups of fixed coclass, advancing understanding of their algebraic structure.

## Key findings

- Finite p-groups of fixed coclass have finitely many mod-p cohomology rings.
- The result is proved by first establishing a stronger version for groups of bounded rank.
- Confirms Carlson's conjecture on cohomology rings of p-groups.

## Abstract

We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of bounded rank.