A note on a class of gyrogroups

TL;DR

This paper investigates the properties of a specific class of gyrogroups, providing conditions for gyrocommutativity and establishing isomorphism criteria related to associated finite groups.

Contribution

It offers a necessary and sufficient condition for gyrocommutativity and characterizes isomorphisms between gyrogroups derived from certain finite groups.

Findings

Characterization of gyrocommutative gyrogroups

Isomorphism criteria for gyrogroups from finite groups

Connection between group properties and associated gyrogroups

Abstract

In \cite{1}, a class of gyrogroups $(G,\odot)$ has been associated to certain groups $(G,\cdot)$. We give a necessary and sufficient condition for $(G,\odot)$ to be gyrocommutative. We also prove that under a suitable assumption two finite groups central by a $2$-Engel group are isomorphic if and only if their associated gyrogroups are isomorphic.