Seeing the Möbius disc-transformation group like never before

The introduction of the gyration notion into nonassociative algebra, hyperbolic geometry, and relatively physics is motivated in this article by the emergence of the gyrogroup notion in the theory of the Möbius transformation group of the complex open unit disc. It suggests the prefix “gyro” that we use to emphasize analogies. Thus, for instance, gyrogroups are classified into gyrocommutative and nongyrocommutative gyrogroups in full analogy with the classification of groups into commutative and noncommutative groups. The road from the Thomas precession of the special theory of relativity to the Thomas gyration as well as the resulting new theory is presented in the authorʹs book: Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1]. The main result of this article is a theorem that allows the validity of some gyration identities to be extended from gyrocommutative gyrogroups into gyrogroups that need not be gyrocommutative. To set the stage for the main result, the Möbius disc-transformation group is studied in a novel way that suggests the notion of the gyrogroup and its gyrations.