Automorphism Groups of Quandles

Abstract

We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [9], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [4]. Since computations of automorphisms of quandles relates to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.

Equivalence classes of colorings

“…In [4] we find the following facts. For a given integer m ≥ 3, each coloring automorphism of Z m is given by: f λ,µ (x) = λx + µ with µ ∈ Z m and λ ∈ Z * m , the set of units of Z m .…”

Enumeration of racks and quandles up to isomorphism

“…This must be the case because the cycle structure of permutations is preserved by conjugation. To give a detailed calculation in one case, there is an edge from (5,6) in C G (G 4 ) to (2,3) in C G (G 1 ) since with x = 1, y = xf −1 = 5 and g y = (4, 5) we have ((5, 6) gy ) f = (2, 3).…”

“…There is a unique non-solvable group among the 471 permutation groups, namely a copy of S 5 generated by (1,4,6,3,2) (5,8,7,10,9) and (3,8) (5,9) (6,10). There are 2 quandles associated with this group.…”

Ternary and n-ary f-distributive structures

“…-For any group G and any group endomorphism f of G, the operation x * y = y − xf (y) de nes a f -quandle structure on G. -Consider the Dihedral quandle Rn, where n ≥ , and let f be an automorphism of Rn. Then f is given by f (x) = ax + b, for some invertible element a ∈ Zn and some b ∈ Zn [16]. The binary operation x * y = f ( y − x) = ay − ax + b (mod n) gives a f -quandle structure called the Dihedral f -quandle.…”

“…We also study extensions and modules, as well as cohomology theory of these structures. For classical quandles theory, we refer to [10], see also [11][12][13][14][15][16][17][18][19]. For basics and some developments of Hom-type algebras, we refer to [8,[20][21][22][23][24][25][26].…”