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.
