Amirkabir University of Technology AUT Journal of Mathematics and Computing 2783-2449 5 1 2024 01 01 Classification of gyrogroups of orders at most 31 11 18 5063 10.22060/ajmc.2023.21939.1125 EN Ali Reza Ashrafi Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran Kurosh Mavaddat Nezhaad Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran Mohammad Ali Salahshour Department of Mathematics, Savadkooh Branch, Islamic Azad University, Savadkooh, Iran Journal Article 2022 11 15 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. https://ajmc.aut.ac.ir/article_5063_67d9057f7f7fb934e9000aefe2393820.pdf