Sylow Theory for Quasigroups

Abstract

This 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 P∖Q, and also in terms of the behavior of the isomorphism type [P∖Q] within the so‐called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.

Sylow Theory for Quasigroups II: Sectional Action

“…Section provides a classification of the divisors d of the order of a finite quasigroup based on the behavior of the pseudo‐orbits of subsets of size d . This classification is correlated with the classification from , which was based on the behavior of orbits under the full left multiplication group.…”

“…This brief section relates the topics of the current section with the concepts of overlapping and nonoverlapping orbits introduced earlier [, §5]. Recall that in a finite quasigroup Q , the orbit of a subset S under the left multiplication group is said to be overlapping if it contains distinct elements that are not disjoint, and otherwise is described as nonoverlapping .…”

“…In general, a left sectional subquasigroup of a finite quasigroup need not be left Lagrangean. Recall that a subquasigroup P of a finite quasigroup Q is left Lagrangean if the relative right multiplication group of P in Q acts semitransitively, so that its orbits all have the same size [, §4.5] [, §4]. Example Consider the opposite Q of the quasigroup of integers modulo 3 under subtraction, with .…”

“…General approaches to the extension have been both top‐down and bottom‐up. The top‐down approach works with the Burnside order, a labeled order structure on the set of isomorphism classes of irreducible permutation representations of a quasigroup [, pp. 10–13].…”

“…The bottom‐up approach, which is modeled on Wielandt's treatment of the Sylow theory for groups , studies the combinatorial or geometric structure of systems of subsets of a quasigroup. In a predecessor paper, the systems studied were the orbits of a given subset under the full left multiplication group of the quasigroup [, pp. 5–9].…”