Construction of All Gyrogroups of Orders at most 31

TL;DR

This paper classifies all gyrogroups of orders up to 31, focusing on small orders, and explores their algebraic structure, showing many are actually groups, with some specific classifications for orders like 8, 12, 15, etc.

Contribution

It provides a complete classification of gyrogroups of orders at most 31, expanding understanding of their structure and relation to groups.

Findings

Gyrogroups of orders p, 2p, p^2 are groups

Classified gyrogroups of orders 8, 12, 15, 18, 20, 21, 28

Many small-order gyrogroups are actually groups

Abstract

The gyrogroup is the closest algebraic structure to the group ever discovered. It has a binary operation $\star$ containing an identity element such that each element has an inverse. Furthermore, for each pair $(a,b)$ of elements of this structure there exists an automorphism $\gyr{a,b}{}$ with this property that left associativity and left loop property are satisfied. Since each gyrogroup is a left Bol loop, some results of Burn imply that all gyrogroups of orders $p, 2p$ and $p^2$ are groups. The aim of this paper is to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.