On the extension problem for gyrogroups

A gyrogroup is an algebraic structure whose operation is in \mbox{general} non-associative and shares common properties with groups. In this paper, we introduce two disjoint families of gyrogroups. One family consists of \mbox{gyrogroups} whose operations are, in some sense, most far from being associative called contra-associative gyrogroups. The other family consists of gyrogroups that are, in some sense, most close to groups called g-extensive gyrogroups. We then describe their structural properties, which eventually lead to studying the extension problem for gyrogoups in detail using the notion of associators. In particular, we refine the hierarchy of gyrogroup structure by showing that generic gyrogroups are extensions of contra-associative gyrogroups or g-extensive gyrogroups.