In this era of an increased interest in loop theory, the Einstein velocity addition law has fresh resonance. One of the most fascinating aspects of recent work in Einstein's special theory of relativity is the emergence of special grouplike loops. The special grouplike loops, known as gyrocommutative gyrogroups, have thrust the Einstein velocity addition law, which previously has operated mostly in the shadows, into the spotlight. We will find that Einstein (Möbius) addition is a gyrocommutative gyrogroup operation that forms the setting for the Beltrami-Klein (Poincaré) ball model of hyperbolic geometry just as the common vector addition is a commutative group operation that forms the setting for the standard model of Euclidean geometry. The resulting analogies to which the grouplike loops give rise lead us to new results in (i) hyperbolic geometry; (ii) relativistic physics; and (iii) quantum information and computation.2000 Mathematics Subject Classification: 20N05 (51P05, 83A05)

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M a t h e m a t i c s phenomena and discovers the secret analogies which unite them. Joseph Fourier. Where there is physical significance, there is pattern and mathematical regularity. The aim of this article is to expose a hitherto unsuspected grouplike structure underlying the set of all relativistically admissible velocities, which shares remarkable analogies with the ordinary group structure. The physical phenomenon that stores the mathematical regularity in the set of all relativistically admissible three-velocities turns out to be the Thomas precession of special relativity theory. The set of all three-velocities forms a group under velocity addition. In contrast, the set of all relativistically admissible three-velocities does not form a group under relativistic velocity addition. Since groups measure symmetry and exhibit mathematical regularity it seems that the progress from velocities to relativistically admissible ones involves a loss of symmetry and mathematical regularity. This article reveals that the lost symmetry and mathematical regularity is concealed in the Thomas precession. Following a presentation of the group axioms, analogous axioms underlying the grouplike structure of velocities in the relativistic regime are presented. These turn out to include the usual group axioms in which the associative–commutative laws are relaxed by means of the Thomas precession. In order to expose the physics student to the power and elegance of abstract mathematics, our results are placed in the context of an abstract real inner product space. However, not much is lost if the student always assumes that the abstract real inner product space is the familiar Euclidean three-space.

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