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

Teerapong Suksumran ()
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Teerapong Suksumran: Chulalongkorn University, Faculty of Science, Department of Mathematics and Computer Science

A chapter in Essays in Mathematics and its Applications, 2016, pp 369-437 from Springer

Abstract: 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.

Keywords: Clifford Algebra; Isomorphism Theorem; Quadratic Space; Left Coset; Moufang Loop (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-31338-2_15

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DOI: 10.1007/978-3-319-31338-2_15

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