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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups and groups implies the existence of a general theory which underlies the theories of groups and gyrogroups and unifies them with respect to their central features. Accordingly, our goal is to construct finite and infinite gyrogroups, both gyrocommutative and non-gyrocommutaive, in order to demonstrate that gyrogroups abound in group theory of which they form an integral part. A gyrogroup is a grouplike structure that is defined in [7] along with a weaker structure called a left gyrogroup. We show that any given group can be turned into a left gyrogroup which is, in turn, a gyrogroup if and only if the given group is central by a 2-Engel group. The importance of left gyrogroups stems from the facts that (i) any gyrotransversal groupoid is a left gyrogroup, and (ii) any left gyrogroup is a twisted subgroup in a specified group. Twisted subgroups are subsets of groups, introduced by Aschbacher [1], which under general conditions are near subgroups. The concept of near subgroup of a finite group was introduced by Feder and Vardi [6 ]a s at ool to study problems in computational complexity involving the class NP .

Gyrogroups and the decomposition of groups into twisted subgroups and subgroups