Modules over semisymmetric quasigroups

TL;DR

This paper characterizes modules over semisymmetric quasigroups, showing their universal multiplication groups are free and describing their stabilizers, leading to new insights into their module theory and extensions.

Contribution

It establishes that the universal multiplication group of a semisymmetric quasigroup is free and details the structure of modules over these quasigroups, including explicit examples.

Findings

Universal multiplication group is free over the underlying set.

Point-stabilizers of the group action are specified.

Examples of semisymmetric quasigroup extensions are provided.

Abstract

The class of semisymmetric quasigroups is determined by the identity $(yx)y=x.$ We prove that the universal multiplication group of a semisymmetric quasigroup $Q$ is free over its underlying set and then specify the point-stabilizers of an action of this free group on $Q$. A theorem of Smith indicates that Beck modules over semisymmetric quasigroups are equivalent to modules over a quotient of the integral group algebra of this stabilizer. Implementing our description of the quotient ring, we provide some examples of semisymmetric quasigroup extensions. Along the way, we provide an exposition of the quasigroup module theory in more general settings.