Compatibility in certain quasigroup homogeneous space. (Q2857009)

The authors consider in this paper a special double-cover \(Q\) of the symmetric group of degree 3. They show that a proper non-regular approximate symmetry occurs from its quasigroup homogeneous space. They consider a quasigroup obtained by permuting entries in the multiplication table of the direct product \(S_3\times Z_2\) of the symmetric group \(S_3\) with the 2-element additive group \(Z_2\) of integers modulo 2, as in [\textit{B. Im} et al., J. Algebr. Comb. 33, No. 1, 81-93 (2011; Zbl 1214.20062)], where the authors consider the general development of the theory of quasigroup permutation representations. In the present paper they want to establish a non-regular approximate symmetry from a quasigroup homogeneous space of degree 4. In Theorem 3.3, p. 673, the authors completely characterize the weak compatibility of any two elements of quasigroup \(Q\). However, the strong compatibility of \(Q\) still remains open. But the authors expect to characterize the compatibility completely for the case of degree 6, in which case the homogeneous space is uniform. For details see [loc. cit.].