Mathematicsphenomenaanddiscoversthesecretanalogieswhichunitethem.JosephFourier. 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.