Gyrogroups associated with groups

TL;DR

This paper investigates the properties of gyrogroups derived from groups of nilpotency class 3, focusing on how their structure relates to the original group, especially regarding nilpotency class and inner mappings.

Contribution

It establishes conditions under which the nilpotency class of the associated gyrogroup matches that of the original group, and explores abelian inner mapping groups in this setting.

Findings

Nilpotency class of gyrogroup equals that of the group if 3 does not divide the group's order.

Studied the structure of abelian inner mapping groups within gyrogroups.

Provided new insights into the algebraic properties of gyrogroups related to groups of nilpotency class 3.

Abstract

In this paper, we study the properties of the associated gyrogroup ${^\circ}G$ of a given group $G$ of nilpotency class $3$. We have proved that if $3$ does not divide the order of the group $G$, then the nilpotency class of the associated gyrogroup ${^\circ}G$ is same as that of the group $G$. We have also studied the problem of abelian inner mapping group in this context.