# Symmetric cohomology of groups

## TL;DR

This paper explores the relationships between symmetric, exterior, and classical group cohomologies, introducing spectral sequences and identifying conditions under which they are isomorphic, such as for torsion-free groups.

## Contribution

It establishes a detailed connection between different cohomology theories of groups, including the introduction of spectral sequences to analyze their relationships.

## Key findings

- Map from exterior to symmetric cohomology is a split monomorphism
- Symmetric and classical cohomologies are isomorphic for torsion-free groups
- Spectral sequences elucidate the relationships between the cohomology theories

## Abstract

We investigate the relationship between the symmetric, exterior and classical cohomologies of groups. The first two theories were introduced respectively by Staic and Zarelua. We show in particular, that there is a map from exterior cohomology to symmetric cohomology which is a split monomorphism in general and an isomorphism in many cases, but not always. We introduce two spectral sequences which help to explain the realtionship between these cohomology groups. As a sample application we obtain that symmetric and classical cohomologies are isomorphic for torsion free groups.