Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

Crossed modules and symmetric cohomology of groups

Pages: 123 – 134

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a7


Mariam Pirashvili (University of Southampton, United Kingdom)


This paper links the third symmetric cohomology (introduced by Staic [10] and Zarelua [12]) 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.


group cohomology, crossed modules, symmetric cohomology

2010 Mathematics Subject Classification

18D05, 20J06

Copyright © 2020, Mariam Pirashvili. Permission to copy for private use granted.

This research was supported by the EPSRC grant EP/N014189/1 “Joining the dots: from data to insight”.

Received 1 April 2019

Received revised 25 July 2019

Accepted 28 August 2019

Published 15 April 2020