Sylow properties of a class of quasigroups (Q1059719)

A quasigroup is called left-distributive if \(x(yz)=(xy)(xz)\) for all x, y, and z. The author shows that, although the group-theoretic Sylow theorems are not true in all finite quasigroups, the situation in left- distributive quasigroups is somewhat better. In a finite solvable left- distributive quasigroup G there exist Hall subquasigroups, provided that the order of a left translation \(L_ a: x\mapsto ax\) is relatively prime to the order of G. This result is then used to prove the solvability of p-quasigroups with the same condition on \(L_ a\). Finally, the author describes an example of a left-distributive quasigroup in which not all Sylow subquasigroups exist.