Gyrogroups and the Cauchy property

Abstract

A gyrogroup is a nonassociative group-like structure. In this article, we extend the Cauchy property from groups to gyrogroups. The (weak) Cauchy property for finite gyrogroups states that if p is a prime dividing the order of a gyrogroup G, then G contains an element of order p. An application of a result in loop theory shows that gyrogroups of odd order as well as solvable gyrogroups satisfy the Cauchy property. Although gyrogroups of even order need not satisfy the Cauchy property, we prove that every gyrogroup of even order contains an element of order two. As an application, we prove that every group of order nq, where n ⊂ N and q is a prime with n < q, contains a unique characteristic subgroup of order q.