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

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Essays in Mathematics and its Applications

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.

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Acknowledgements

I wish to thank my advisor Keng Wiboonton for his orientation, guidance, and encouragement during this work. I would like to thank Professor Themistocles M. Rassias and Professor Abraham A. Ungar for the kind invitation to contribute a paper to this volume dedicated to the memory of Vladimir Arnold and for their helpful suggestions during the preparation of the manuscript. This work was partially supported by National Science Technology Development Agency (NSTDA), through the Junior Science Talent Project (JSTP). The financial support from the Institute for the Promotion of Teaching Science and Technology (IPST), through the Development and Promotion of Science and Technology Talents Project (DPST), is highly appreciated.

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  1. Faculty of Science, Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok, 10330, Thailand

    Teerapong Suksumran

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  1. Teerapong Suksumran
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Correspondence to Teerapong Suksumran .

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  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. University of Florida, GAINESVILLE, Florida, USA

    Panos M. Pardalos

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Suksumran, T. (2016). The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_15

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