Sur l'application de Quillen pour la cohomologie modulo 2 de certains groupes finis

TL;DR

This paper demonstrates that Quillen's isomorphism for mod 2 cohomology of finite groups is actually an isomorphism for symmetric, alternating, and finite Coxeter groups, confirming its broader applicability.

Contribution

The paper extends Quillen's isomorphism to specific families of finite groups, showing it is an actual isomorphism beyond the original cases.

Findings

Quillen's map is an isomorphism for symmetric groups

Quillen's map is an isomorphism for alternating groups

Quillen's map is an isomorphism for finite Coxeter groups

Abstract

Let $G$ be a finite group. In a famous article, Quillen describes an $\mathrm{F}$-isomorphism between commutative $\mathbb{N}$-graded $\mathbb{F}_{2}$-algebras $$\mathrm{q}_{G}:\mathrm{H}^{*}(G;\mathbb{F}_{2})\to\mathrm{L}(G)\ ,$$ with $\mathrm{L}(G)$ defined in terms of the elementary abelian $2$-groups contained in $G$. We show that $\mathrm{q}_{G}$ is in fact an actual isomorphism for three families of finite groups: symmetric groups, alternating groups and finite Coxeter groups.