# Crossed modules and symmetric cohomology of groups

## TL;DR

This paper explores the relationship between third symmetric cohomology of groups and crossed modules, revealing conditions under which group extensions correspond to elements of symmetric cohomology, with implications for 2-group structures.

## Contribution

It establishes a connection between symmetric cohomology and crossed modules, extending known results to 2-groups and clarifying the role of inverse-preserving sections.

## Key findings

- Extension of 2-groups corresponds to symmetric cohomology elements with inverse-preserving sections
- Links symmetric cohomology to properties of crossed modules and 2-groups
- Generalizes Staic and Zarelua's results on symmetric cohomology and abelian extensions

## Abstract

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.