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

In: Essays in Mathematics and its Applications

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  • Teerapong Suksumran

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

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.

Suggested Citation

  • Teerapong Suksumran, 2016. "The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems," Springer Books, in: Themistocles M. Rassias & Panos M. Pardalos (ed.), Essays in Mathematics and its Applications, pages 369-437, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31338-2_15
    DOI: 10.1007/978-3-319-31338-2_15
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