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.

Abstract. Considering a special double-cover Q of the symmetric group of degree 3, we show that a proper non-regular approximate symmetry occurs from its quasigroup homogeneous space. The weak compatibility of any two elements of Q is completely characterized in any such quasigroup homogeneous space of degree 4.