Special subgroups of gyrogroups: Commutators, nuclei and radical

TL;DR

This paper explores the internal subgroup structures of gyrogroups, including commutators, nuclei, and radicals, providing criteria for normality in finite cases, thus advancing the algebraic understanding of these nonassociative structures.

Contribution

It introduces new subgroup concepts within gyrogroups and establishes conditions for their normality, enhancing the algebraic theory of gyrogroups.

Findings

Normal closures of key subgyrogroups are normal subgroups.

Criteria for normality of subgyrogroups in finite gyrogroups.

Identification of specific subgroup structures like commutator, nucleus, and radical.

Abstract

A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein's velocity addition law. In this article, we present a few of groups sitting inside a gyrogroup $G$, including the commutator subgyrogroup, the left nucleus, and the radical of $G$. The normal closure of the commutator subgyrogroup, the left nucleus, and the radical of $G$ are in particular normal subgroups of $G$. We then give a criterion to determine when a subgyrogroup $H$ of a finite gyrogroup $G$, where the index $[G\colon H]$ is the smallest prime dividing $|G|$, is normal in $G$.