Normal Subgyrogroups of Certain Gyrogroups

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.