Associativization of gyrogroups and the universal property

A gyrogroup is a non-associative structure that may be regarded as a suitable generalization of groups. In this paper, we introduce the notion of an associator in an arbitrary gyrogroup that, in some sense, measures the deviation from associativity of gyrogroup operations. We then study the normal closure of the set of associators and establish several relevant properties, including the universal property of the associativization of a gyrogroup. This leads to a few invariant properties of gyrogroups in terms of associator normal subgyrogroups and gives an effective method to study representations of gyrogroups.