Normal Subgyrogroups of Certain Gyrogroups

TL;DR

This paper investigates the structure of normal subgyrogroups within certain gyrogroups, expanding understanding of their algebraic properties and potential classifications.

Contribution

It provides new insights into the properties and classifications of normal subgyrogroups in specific gyrogroups, a topic not extensively explored before.

Findings

Characterization of normal subgyrogroups in certain gyrogroups

Conditions under which subgyrogroups are normal

Structural properties of these subgyrogroups

Abstract

Suppose that $(T,\star)$ is a groupoid with a left identity such that each element $a\in T$ has a left inverse. Then $T$ is called a \textit{gyrogroup} if and only if $(i)$ there exists a function $gyr:T\times T\longrightarrow Aut(T)$ such that for all $a,b,c\in T$, $a\star(b\star c)= (a\star b)\star gyr[a,b]c$, where $gyr[a,b]c=gyr(a,b)(c)$; and $(ii)$ for all $a,b\in T$, $gyr[a,b]=gyr[a\star b,b]$. In this paper, the structure of normal subgyrogroups of certain gyrogroups are investigated.