Sylow Theory for Quasigroups

Abstract

AbstractThis paper is intended as a first step toward a general Sylow theory for quasigroups and Latin squares. A subset of a quasigroup lies in a nonoverlapping orbit if its respective translates under the elements of the left multiplication group remain disjoint. In the group case, each nonoverlapping orbit contains a subgroup, and Sylow's Theorem guarantees nonoverlapping orbits on subsets whose order is a prime‐power divisor of the group order. For the general quasigroup case, the paper investigates the relationship between non‐overlapping orbits and structural properties of a quasigroup. Divisors of the order of a finite quasigroup are classified by the behavior of nonoverlapping orbits. In a dual direction, Sylow properties of a subquasigroup P of a finite left quasigroup Q may be defined directly in terms of the homogeneous space , and also in terms of the behavior of the isomorphism type within the so‐called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.