Inverse property of non-associative abelian extensions

TL;DR

This paper characterizes linear abelian extensions of commutative groups by loops, focusing on their properties and applications to ordered loops and finite structures, using an equivariant group action framework.

Contribution

It provides a constructive characterization of linear abelian extensions with various properties based on an equivariant group action approach.

Findings

Extensions can be simplified for ordered loops

Characterization of extensions with inverse properties

Determination of possible cardinalities of component loops

Abstract

Our paper deals with the investigation of extensions of commutative groups by loops so that the quasigroups that result in the multiplication between cosets of the kernel subgroup are T-quasigroups. We limit our study to extensions in which the quasigroups determining the multiplication are linear functions without constant term, called linear abelian extensions. We characterize constructively such extensions with left-, right-, or inverse properties using a general construction according to an equivariant group action principle. We show that the obtained constructions can be simplified for ordered loops. Finally, we apply our characterization to determine the possible cardinalities of the component loop of finite linear abelian extensions.