Gyrogroup actions: A generalization of group actions

Abstract

This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three well-known theorems in group theory for gyrogroups: the orbit-stabilizer theorem, the orbit decomposition theorem, and the Burnside lemma (or the Cauchy-Frobenius lemma). We then prove that under a certain condition, a gyrogroup G acts transitively on the set G/H of left cosets of a subgyrogroup H in G in a natural way. From this we prove the structure theorem that every transitive action of a gyrogroup can be realized as a gyrogroup action by left gyroaddition. We also exhibit concrete examples of gyrogroup actions from the Möbius and Einstein gyrogroups.

Left Regular Representation of Gyrogroups

“…for all a, b ∈ G. By Theorems 3.2 and 3.3 of [1], the notions of gyrogroup actions and permutation representations are equivalent. In fact, if • is an action of G on X, then the map ϕ :…”

“…Gyrogroups are nonassociative algebraic structures that share many properties with groups [1,2,4,6,7]. One of the main aspects of gyrogroup structures is that the Cayley table of a finite gyrogroup represents a 'Latin square'-an object studied in combinatorics and in experimental design.…”

“…This follows from the fact that the equations a ⊕ x = b and x ⊕ a = b in the variable x always have solutions in gyrogroups; see, for instance, Theorem 2.22 of [7]. Important combinatorial results such as the orbit-stabilizer theorem and the Cauchy-Frobenius lemma remain valid in the case of gyrogroups [1], among other things.…”

“…In [1], we introduce a generalization of group actions. By an action of a gyrogroup G on a nonempty set X we mean a map…”

“…Hence, it seems difficult to extend the notion of group actions to nonassociative structures. However, this can be done (in certain circumstances) by employing the equivalence of group actions and permutation representations, see, for instance, [1][2][3][4]. This motivates us to study the left regular representation in the setting of gyrogroups in more detail.…”