Gyrogroups and the decomposition of groups into twisted subgroups and subgroups

Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein's velocity addit ion as a binary operation. Einstein's gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ion does not result in loss of mathematical regularity owing to the presenc e of the relativistic effect known as the Thomas precession which, by abstr action, becomes an automorphism called the Thomas gyration. The Thomas gyra tion turns out to be the missing link that gives rise to analogies shared b y gyrogroups and groups. In particular, it gives rise to the gyroassociativ e and the gyrocommuttive laws that Einstein's addition possesses, in full a nalogy with the associative and the commutative laws that vector addition p ossesses in a vector space. The existence of striking analogies shared by g yrogroups and groups implies the existence of a general theory which underl ies the theories of groups and gyrogroups and uni es them with respect to t heir central features. Accordingly, our goal is to construct finite and inf inite gyrogroups, both gyrocommutative and non-gyrocommutaive, in order to demonstrate that gyrogroups abound in group theory of which they form an in tegral part.