Classification of gyrogroups of orders at most 31

Document Type : Original Article


1 Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran

2 Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran


A gyrogroup is defined as having 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 ${\mathrm{gyr}}[a,b]$ 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 $p^2$, where $p$ is a prime number, are groups. This paper aims to classify gyrogroups of orders 8, 12, 15, 18, 20, 21, and 28.


Main Subjects

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