Sharply transitive sets in quasigroup actions

Abstract

This paper forms part of the general development of the theory of quasigroup permutation representations. Here, the concept of sharp transitivity is extended from group actions to quasigroup actions. Examples of nontrivial sharply transitive sets of quasigroup actions are constructed. A general theorem shows that uniformity of the action is necessary for the existence of a sharply transitive set. The concept of sharp transitivity is related to two pairwise compatibility relations and to maximal cliques within the corresponding compatibility graphs.

Gyrogroup actions: A generalization of group actions

“…For instance, Jonathan Smith has intensively studied quasigroup and loop representations in a series of papers [9][10][11]. Also, the study of sharply transitive sets in quasigroup actions can be found in [7].…”

Compatibility in Certain Quasigroup Homogeneous Space

“…We consider a quasigroup obtained by permuting entries in the multiplication table of the direct product S 3 × Z 2 of the symmetric group S 3 with the 2-element additive group Z 2 of integers modulo 2 as in [3], and want to establish a non-regular approximate symmetry from a quasigroup homogeneous space of degree 4 in this paper.…”

“…Six elements of S 3 are denoted as three rotations ρ 0 = (0), ρ 1 = (021), ρ 2 = (012), and three reflections σ 0 = (12), σ 1 = (02), σ 2 = (01). Z 2 = {0, 1} as in [3]. Consider a relation λ · µ = ν in S 3 .…”

“…And such graph-theoretical characterization of compatibility will be used to study sharp tnansitivity in a quasigroup. For further details, see [3].…”