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

DOI10.1007/978-3-319-31338-2_15zbMath1402.20070OpenAlexW2498476507MaRDI QIDQ4554334

Teerapong Suksumran

Publication date: 13 November 2018

Published in: Essays in Mathematics and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-3-319-31338-2_15

zbMATH Keywords

Clifford algebraMoufang loopisomorphism theoremquadratic spaceleft coset

Mathematics Subject Classification ID

Loops, quasigroups (20N05) Clifford algebras, spinors (15A66)

Related Items (29)

Möbius's functional equation and Schur's lemma with applications to the complex unit disk ⋮ Quotient with respect to admissible \(L\)-subgyrogroups ⋮ Cardinal invariants in the quotient spaces of weakly topological groups ⋮ Geometry of gyrogroups via Klein's approach ⋮ Unnamed Item ⋮ On Schreier graphs of gyrogroup actions ⋮ Fuzzy gyronorms on gyrogroups ⋮ On commutator-inversion invariant groups and gyrogroup construction ⋮ The Hartman-Mycielski construction in topological gyrogroups ⋮ The first-countability in the quotient spaces of topological gyrogroups ⋮ Unnamed Item ⋮ Unnamed Item ⋮ On Metric Structures of Normed Gyrogroups ⋮ The Isometry Group of n-Dimensional Einstein Gyrogroup ⋮ An Inequality Related to Möbius Transformations ⋮ Construction of New Gyrogroups and the Structure of their Subgyrogroups ⋮ Topological gyrogroups: generalization of topological groups ⋮ Involutive groups, unique 2-divisibility, and related gyrogroup structures ⋮ A note on paratopological loops ⋮ Extension of the Švarc-Milnor lemma to gyrogroups ⋮ Complete regularity of paratopological gyrogroups ⋮ Complete reducibility of gyrogroup representations ⋮ Embedding of locally compact Hausdorff topological gyrogroups in topological groups ⋮ Extension of Maschke’s theorem ⋮ A note on (strongly) topological gyrogroups ⋮ On paratopological gyrogroups ⋮ Feathered gyrogroups and gyrogroups with countable pseudocharacter ⋮ Extensions of some topological properties in topological gyrogroups ⋮ Extended Cartan fixed point theorem

Cites Work

• Unit balls, Lorentz boosts, and hyperbolic geometry
• Complex boosts: a Hermitian Clifford algebra approach
• Hyperbolic geometry
• Harmonic analysis on the Einstein gyrogroup
• Möbius gyrogroups: a Clifford algebra approach.
• The finite Bruck loops.
• On Bol loops of exponent 2.
• The relativistic noncommutative nonassociative group of velocities and the Thomas rotation
• Factorizations of invertible density matrices
• Factorizations of Möbius gyrogroups.
• Einstein gyrogroup as a B-loop
• On algebraic structures related to Beltrami-Klein model of hyperbolic geometry
• Spherical continuous wavelet transforms arising from sections of the Lorentz group
• K-loops, gyrogroups and symmetric spaces
• On the notion of gyrogroup
• Loops and the Lagrange property.
• Theory of K-loops
• Symmetric sets with midpoints and algebraically equivalent theories
• Physical applications of homogeneous balls. With the assistance of Tzvi Scarr
• Strong near subgroups and left gyrogroups
• Defect and area in Beltrami-Klein model of hyperbolic geometry
• The cogyrolines of Möbius gyrovector spaces are metric but not periodic
• The Einstein relativistic velocity model of hyperbolic geometry and its plane separation axiom
• Gyrogroups and homogeneous loops
• On trigonometry in Beltrami-Klein model of hyperbolic geometry
• Isomorphism theorems for gyrogroups and L-subgyrogroups.
• Einstein's velocity addition law and its hyperbolic geometry
• The proper-time Lorentz group demystified
• On twisted subgroups and Bol loops of odd order.
• Thomas rotation and Thomas precession
• A History of Lagrange's Theorem on Groups
• The Classification of Groups of Order 2p
• A Characterization of Infinite Cyclic Groups
• Topological Right Gyrogroups and Gyrotransversals
• Distances of qubit density matrices on Bloch sphere
• A class of finite simple Bol loops of exponent 2
• From Möbius to Gyrogroups
• A Gyrovector Space Approach to Hyperbolic Geometry
• Finite Bol loops
• On Groups of Order p²
• Loops and semidirect products
• Moufang Loops of Small Order. I
• On Groups of Order pq
• UNIQUELY 2-DIVISIBLE BOL LOOPS
• Involutory decomposition of groups into twisted subgroups and subgroups
• Lagrange's Theorem for Gyrogroups and the Cauchy Property
• Gyrations: The Missing Link Between Classical Mechanics with Its Underlying Euclidean Geometry and Relativistic Mechanics with Its Underlying Hyperbolic Geometry
• Abstract space–times and their Lorentz groups
• Lagrange's theorem for Moufang loops
• Gyrogroups in Projective Hyperbolic Clifford Analysis
• Gyrogroups and the decomposition of groups into twisted subgroups and subgroups.
• Beyond the Einstein addition law and its gyroscopic Thomas precession. The theory of gyrogroups and gyrovector spaces
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item

This page was built for publication: The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems