On the Orbit Decomposition of Finite Quandles

Abstract

We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe a procedure for finding all non-connected quandle structures on a disjoint union of subquandles.

Toward a Classification of Finite Quandles

“…a Shortly after the conclusion of the KSU REU, Nelson and Wong[6] announced the independent discovery of a decomposition theorem equivalent to this theorem and the preceding, when viewed as a decomposition theorem, rather than a construction. In their result, the extra structure is expressed in terms of compatible rack actions, rather than compatible group homomorphisms from universal augmentation groups to automorphism groups.J.…”

A Polynomial Invariant of Finite Quandles

“…The qp polynomial distinguishes between some quandles which have the same orbit decomposition but different structure maps (see [11]).…”

Quandles and Linking Number

“…A quandle Q is decomposable if we may write Q = Q 1 ∪ Q 2 where Q 1 ∩ Q 2 = ∅ and Q 1 , Q 2 are both subquandles of Q. Note that even if a quandle is not decomposable in this sense, we may still be able to write Q = n i=1 Q i for Q i disjoint subquandles; see [1] and [8]. A quandle is connected if there is an element a ∈ Q such that every element of Q is equivalent to a word of the form…”