Symmetric cohomology of groups in low dimension

TL;DR

This paper characterizes group extensions related to symmetric cohomology in low dimensions and explores conditions for the injectivity of the natural map to ordinary cohomology.

Contribution

It provides explicit criteria for group extensions in symmetric cohomology and analyzes the injectivity of the comparison map to standard cohomology.

Findings

Explicit characterization of group extensions in symmetric cohomology

Conditions for injectivity of the map to ordinary cohomology

Insights into low-dimensional symmetric cohomology structure

Abstract

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology $HS^2(G,A)$. We also give conditions for the map $HS^n(G,A)\to H^n(G,A)$ to be injective.