The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Abstract

Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible gyrocommutative gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Möbius addition. In the second part of this paper, we show that the symmetric group of a gyrogroup admits the gyrogroup structure, thus obtaining an analog of Cayley's theorem for gyrogroups. We examine subgyrogroups, gyrogroup homomorphisms, normal subgyrogroups, and quotient gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange's theorem for gyrogroups and use this result to prove that gyrogroups of particular order have the Cauchy property.

On Metric Structures of Normed Gyrogroups

“…Proof. Suppose that f is an isometry of G. By definition, f is a permutation of G. By Proposition 19 of [22], f = L f (e) • ρ, where ρ is a permutation of G that fixes e. As in the proof of Theorem 18 of [22], L −1 f (e) = L f (e) and so ρ = L f (e) • f . Hence, ρ is an isometry of G, being the composite of isometries of G.…”

“…The table below summarizes some algebraic properties of gyrogroups [22,28], which will prove useful in studying topological and geometric aspects of gyrogroups in Sections 3 and 4. We remark that gyroautomorphisms play an essential role in gyrogroup theory; for example, they appear as part of generic algebraic rules extended from group-theoretic identities.…”

“…Proof. Let a ∈ G. Recall that the left gyrotranslation by a, denoted by L a , is defined by L a (x) = a ⊕ x for all x ∈ G. By Theorem 18 (1) of [22], L a is a bijection from G to itself. Next, we prove that the gyronorm metric d is invariant under L a .…”

Novel Tools to Determine Hyperbolic Triangle Centers

“…Identity (31) results from the following chain of equations, which are numbered for subsequent derivation: To verify (32) we consider the special case of (29) when b D «a, obtaining gyrOEa; «a˚cgyrOE«a; c D gyrOE0; gyrOEa; «acgyrOEa; «a D I (38) where the second identity in (38) follows from items (2) and (3) of Theorem 2.…”

Parametric Realization of the Lorentz Transformation Group in Pseudo-Euclidean Spaces

“…The concept of the gyrogroup emerged from the 1988 study of the parametrization of the Lorentz group in [18]. Presently, the gyrogroup concept plays a universal computational role, which extends far beyond the domain of special relativity, as noted by Chatelin in [1, p. 523] and in references therein and as evidenced, for instance, from [2,4,5,6,7,11,14,15,16] and [12,21,26,28]. In a similar way, the concept of the bi-gyrogroup emerges in this paper from the study of the parametrization of the Lorentz group SO(m, n), m, n ∈ N. Hence, like gyrogroups, bi-gyrogroups are capable of playing a universal computational role that extends far beyond the domain of Lorentz transformations in pseudo-Euclidean spaces.…”