# Mod-two cohomology rings of alternating groups

## TL;DR

This paper computes the mod-two cohomology rings of all alternating groups, revealing their structure, Steenrod algebra action, and nilpotency properties using advanced algebraic topology techniques.

## Contribution

It provides a comprehensive calculation of the mod-two cohomology rings of alternating groups, including cup and transfer products, and analyzes their algebraic properties.

## Key findings

- No nilpotent elements in the cohomology rings of alternating groups
- Explicit description of the Steenrod algebra action
- Determination of the ring and additive structures

## Abstract

We calculate the mod-two cohomology of all alternating groups together, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques is needed: an almost Hopf ring structure associated to the embeddings of products of alternating groups, the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups, Fox-Neuwirth resolutions, and restriction to elementary abelian subgroups.