On universally stable elements

TL;DR

This paper demonstrates that specific subrings of a finite p-group's cohomology can be realized as images of restriction from virtually free groups, leading to finiteness results for these modules.

Contribution

It introduces a method to realize certain cohomology subrings as images from virtually free groups, extending understanding of cohomological structures.

Findings

Cohomology subrings can be realized via restriction from virtually free groups.

The cohomology of P is a finite module over these subrings.

Includes examples like universally stable elements and Dickson algebras.

Abstract

We show that certain subrings of the cohomology of a finite p-group P may be realised as the images of restriction from suitable virtually free groups. We deduce that the cohomology of P is a finite module for any such subring. Examples include the ring of `universally stable elements' defined by Evens and Priddy, and rings of invariants such as the mod-2 Dickson algebras.