Small connected quandles

Abstract

In [6] an attempt was made to classify quandles of small order. There appear to be 404 quandles of order 5, 6658 quandles of order 6, 152900 quandles of order 7 and 5225916 quandles of order 8. For order 9 the search space was too large to finish the computation. If however one restricts to the important subclass of connected quandles then classification seems to be more accessible to computation, comparable to the classification of groups of given order. It is the purpose of this note to classify connected quandles up to order 14, and in particular to show that there is no connected quandle of order 14.

On the Classification of Quandles of Low Order

“…gap> NrSmallQuandles (10); 1 gap> Q := SmallQuandle(10, 1);; gap> R := Rack(SymmetricGroup( 5), (1,2));; gap> IsomorphismRacks(Q, R); (3,5,6,10,8,4,9,7) Recall that a crossed set is a quandle (X, ⊲) which further satisfies j ⊲ i = i whenever i ⊲ j = j for all i, j ∈ X.…”

“…Of course, the classification of finite racks (or quandles) is a very difficult problem. Several papers about classifications of different subcategories of racks have appear, see for example [18], [19], [10], [16], [5], [15], [8].…”

Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles

“…Some quandles on this list are of the form c n λ for some choice of n and λ, although this language is not used in that work. It has also been shown [Cl2,HMN] that conjugation quandles of the symmetric groups are useful in classifying finite quandles.…”

Distributivity in Quandles and Quasigroups

“…However if one restricts himself to the subclass of connected quandles then classification becomes more accessible to calculation in somehow comparable way to the classification of finite groups. In [27], Clauwens studied connected quandles and proved the following Proposition 3.4 [27] If f : Q → P is a surjective quandle homomorphism and P is connected then for all x, y ∈ P , there is a bijection between f −1 (x) and f −1 (y). In particular the cardinality of P divides the cardinality of Q.…”