The prolongation of central extensions

TL;DR

This paper develops an obstruction theory and classification method for the prolongation of central extensions using low-dimensional group cohomology, advancing understanding of their structure and properties.

Contribution

It introduces a new framework for studying $(oldsymbol{ extalpha}, oldsymbol{ extgamma})$-prolongations of central extensions, including classification via cohomology.

Findings

Established an obstruction theory for $(oldsymbol{ extalpha}, oldsymbol{ extgamma})$-prolongations.

Classified such prolongations using low-dimensional cohomology groups.

Provided tools for analyzing the structure of extended central groups.

Abstract

The aim of this paper is to study the $(\alpha, \gamma)$-prolongation of central extensions. We obtain the obstruction theory for $(\alpha, \gamma)$-prolongations and classify $(\alpha, \gamma)$-prolongations thanks to low-dimensional cohomology groups of groups.