Classification of gyrogroups of orders at most 31

Abstract

A gyrogroup is defined as having a binary operation 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 the property that left associativity and the 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 p2, where p is a prime number, are groups. This paper aims to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.

Keywords

• gyrogroup
• left bol loop
• gyroautomorphism