Algebraic properties of gyrogroups

Abstract

In the first part of the dissertation, we give an algebraic proof that the open unit ball B of Euclidean space Rn, equipped with Einstein addition E, forms a uniquely 2-divisible gyrocommutative gyrogroup or, equivalently, a B-loop, using the Clifford algebra formalism. As a consequence, we obtain a compact formula for Einstein addition in terms of Möbius addition. We then give a characterization of associativity and commutativity of vectors in  with respect to Einstein addition. In the second part of the dissertation, we study gyrogroups from an algebraic point of view. We extend some well-known theorems in group theory to gyrogroups, including Cayley’s theorem, the isomorphism theorems, and Lagrange’s theorem. We also prove that gyro-groups of particular order satisfy the Cauchy property.