Construction of gyrogroups of order 2n by cyclic 2-groups

Gyrogroups abound in group theory. They possess a rich group-like structure that forms a natural generalization of groups. A prominent example is provided by Einstein’s addition law of relativistically admissible velocities. Being nonassociative, it turns out that Einstein addition is a nongroup gyrogroup binary operation. A gyrogroup is a rich structure constituting a non-empty set with a binary operation that obeys an associative-like law called the gyroassociative law. The aim of this article is to present a method of constructing novel gyrogroups of order 2n, n > 3, by a cyclic 2-group that is Z2n.