On Schreier graphs of gyrogroup actions |
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| Institution: | Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand;Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai-400 076, India;Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John''s, NL A1C 5S7, Canada;Department of Mathematics, University of Athens, Athens 15784, Greece;Departamento de Matemática/IMAS, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina;Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China;Department of Mathematics, University of Wisconsin, Madison, WI, United States of America |
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| Abstract: | The notion of a group action can be extended to the case of gyrogroups. In this article, we examine a digraph and graph associated with a gyrogroup action on a finite nonempty set, called a Schreier digraph and graph. We show that algebraic properties of gyrogroups and gyrogroup actions such as being gyrocommutative, being transitive, and being fixed-point-free are reflected in their Schreier digraphs and graphs. We also prove graph-theoretic versions of the three fundamental theorems involving actions: the Cauchy–Frobenius lemma (also known as the Burnside lemma), the orbit-stabilizer theorem, and the orbit decomposition theorem. Finally, we make a connection between gyrogroup actions and actions of symmetric groups by evaluation via Schreier digraphs and graphs. |
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| Keywords: | Schreier graph Gyrogroup action Group action Orbit-stabilizer Finite symmetric group |
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