CONNECTED QUANDLES AND TRANSITIVE GROUPS ALEXANDER HULPKE, DAVID STANOVSK ´ Y, AND PETR VOJT ˇ ECHOVSK ´ Y Abstract. Building on ideas of Galkin, we establish a canonical representation of con- nected quandles as certain configurations in transitive groups, called quandle envelopes. This characterization allows us to efficiently enumerate connected quandles of small orders, and present new proofs concerning connected quandles of order p and 2p. We also charac- terize affine connected quandles. 1. Introduction 1.1. A note on terminology. Quandles have been rediscovered in several disguises [2, 3, 22, 28, 31, 32, 42, 44] and the terminology therefore varies greatly. For the most part we keep the modern “quandle” terminology that emerged over the last 10 years. However, in some cases we use the older and more general terminology for binary systems developed to a great extent by R.H. Bruck in his 1958 book [4]. Bruck’s terminology is used fairly consistently in universal algebra, semigroup theory, loop theory and other branches of algebra. For instance, we speak of “right translations” rather than “inner mappings.” 1.2. Racks, quandles and connected quandles. A binary algebra Q =(Q, ·) is called a right quasigroup if all right translations R x : Q → Q, y 7→ yx are permutations of Q. In a right quasigroup, the permutation group RMlt(Q)= hR x : x ∈ Qi is known as the right multiplication group. A right quasigroup satisfying the right distributive law (yz )x =(yx)(zx) is called a rack. Equivalently, a binary algebra Q is a rack if RMlt(Q) is a subgroup of the automorphism group Aut(Q). A rack that satisfies the idempotent law xx = x is called a quandle. A rack Q is said to be connected (also algebraically connected, transitive, homogeneous, indecomposable ) if the natural action of RMlt(Q) is transitive on Q. Every rack decomposes 2000 Mathematics Subject Classification. Primary: 57M27. Secondary: 20N02, 20N05, 20B10. Key words and phrases. Quandle, connected quandle, indecomposable quandle, Galkin representation, enumeration of connected quandles, affine quandle, medial quandle, quandle envelope, rack, connected rack, transitive group of degree 2p. Research partially supported by the Simons Foundation Collaboration Grant 244502 to Alexander Hulpke, the GA ˇ CR grant 13-01832S to David Stanovsk´ y, and the Simons Foundation Collaboration Grant 210176 to Petr Vojtˇ echovsk´ y. 1
Transcript
CONNECTED QUANDLES AND TRANSITIVE GROUPS
ALEXANDER HULPKE, DAVID STANOVSKY, AND PETR VOJTECHOVSKY
Abstract. Building on ideas of Galkin, we establish a canonical representation of con-nected quandles as certain configurations in transitive groups, called quandle envelopes.This characterization allows us to efficiently enumerate connected quandles of small orders,and present new proofs concerning connected quandles of order p and 2p. We also charac-terize affine connected quandles.
1. Introduction
1.1. A note on terminology. Quandles have been rediscovered in several disguises [2, 3,22, 28, 31, 32, 42, 44] and the terminology therefore varies greatly. For the most part we keepthe modern “quandle” terminology that emerged over the last 10 years. However, in somecases we use the older and more general terminology for binary systems developed to a greatextent by R.H. Bruck in his 1958 book [4]. Bruck’s terminology is used fairly consistently inuniversal algebra, semigroup theory, loop theory and other branches of algebra. For instance,we speak of “right translations” rather than “inner mappings.”
1.2. Racks, quandles and connected quandles. A binary algebra Q = (Q, ·) is called aright quasigroup if all right translations
Rx : Q→ Q, y 7→ yx
are permutations of Q. In a right quasigroup, the permutation group
RMlt(Q) = 〈Rx : x ∈ Q〉is known as the right multiplication group. A right quasigroup satisfying the right distributivelaw
(yz)x = (yx)(zx)
is called a rack. Equivalently, a binary algebra Q is a rack if RMlt(Q) is a subgroup of theautomorphism group Aut(Q). A rack that satisfies the idempotent law
xx = x
is called a quandle.A rack Q is said to be connected (also algebraically connected, transitive, homogeneous,
indecomposable) if the natural action of RMlt(Q) is transitive on Q. Every rack decomposes
2000 Mathematics Subject Classification. Primary: 57M27. Secondary: 20N02, 20N05, 20B10.Key words and phrases. Quandle, connected quandle, indecomposable quandle, Galkin representation,
enumeration of connected quandles, affine quandle, medial quandle, quandle envelope, rack, connected rack,transitive group of degree 2p.
Research partially supported by the Simons Foundation Collaboration Grant 244502 to Alexander Hulpke,the GACR grant 13-01832S to David Stanovsky, and the Simons Foundation Collaboration Grant 210176to Petr Vojtechovsky.
1
into orbits of transitivity. The orbits are not necessarily connected, but they share certainproperties with connected racks.
In this paper we are mostly interested in connected quandles, but some of our observationsapply to general quandles and racks as well. In [42, Section 4], one can find hints how tobuild connected racks over connected quandles.
1.3. Motivation and history of connected quandles. One of the main motivationsbehind the theory of quandles is finding computable knot and link invariants—the threedefining properties of quandles correspond to the three Reidemeister moves [22, 32]. Con-nected quandles are of prime interest because all colors used in a knot coloring fall into thesame orbit of transitivity. Disconnected quandles are of importance for links.
From a broader perspective, quandles are a special type of set-theoretical solutions to thequantum Yang-Baxter equation as formulated by Drinfeld [9]. There are indications, see [11],that understanding racks and quandles is an important step towards understanding generalset-theoretical solutions of the quantum Yang-Baxter equation.
The investigation of racks started with several special cases. A rack Q is called involutoryif R2
x = 1 for every x ∈ Q, medial if it satisfies the identity
(xy)(uv) = (xu)(yv),
and latin if all left translations
Lx : Q→ Q, y 7→ xy
are permutations of Q. It is not hard to check that every latin rack is a connected quandle.The theory of latin quandles (also known as right distributive quasigroups) started well
before the term “quandle” appeared. The following are the main structural results for latinquandles:
Medial latin quandles are affine, by the early Toyoda-Bruck theorem [4]. Latin quandlesthat are also left distributive are affine over commutative Moufang loops [2], and many strongproperties follow from this connection; see [26, 41]. General latin quandles were studied ina series of papers by Belousov, Galkin and their collaborators; see Belousov’s book [2] andGalkin’s survey paper [13] for more information. Involutory latin quandles are essentiallythe same objects as Bruck loops of odd order [15, 27].
The theory of quandles is younger, particularly the theory of connected quandles. Here isa brief survey of results on connected quandles related to our work:
A variation on Galkin style representation can be found in [10, 22], where some ideasof [12] were rediscovered. Vendramin [45] used this representation to enumerate connectedquandles up to order n ≤ 35. Different methods lead to the classification of quandles onp, p2 and 2p elements, where p is a prime. Connected quandles of size p and p2 are affine[11, 16]. There are no connected quandles of size 2p, where p is a prime bigger than 5 [33].Simple quandles of size bigger than 2 are connected, and were classified by Joyce [23]. Earlystructural results on connected involutory quandles appeared in [24, 35, 39], and more canbe found in [37, 38] for general connected quandles.
An attempt to understand the orbit decomposition in general quandles was made in [10,36], and stronger results were obtained in certain special cases: for medial quandles see [21],and for involutory quandles see [39].
2
1.4. Summary of results. Our main result, Theorem 4.3, is a characterization of connectedquandles as certain configurations in transitive groups. Some variants of this representationwere discovered independently in [10, 12, 22, 39], but none of these works contains a completecharacterization of our Theorem 4.3, nor a discussion of the isomorphism problem as in ourTheorem 4.6. Using the representation theory, we can prove several known results in asimpler and faster way, and also obtain new results.
The paper is organized as follows. In Section 2 we develop basic properties of quandlesand racks in relation to the right multiplication group and its derived subgroup. Sections 3and 4, which are strongly influenced by work of Galkin, contain a minimal representation(Theorem 3.5) as well as the canonical representation (Theorem 4.3) for connected quandles.We also describe all isomorphisms between connected quandles in canonical representation(Lemma 4.5), and thus solve the isomorphism problem for canonical representations (Theo-rem 4.6), and describe the automorphism group of a connected quandle in terms of its rightmultiplication group (Proposition 4.8).
Section 5 contains a characterization of connected affine quandles (Theorem 5.3). Weshow that connected quandles are affine if and only if they are medial if and only if theirright multiplication group is metabelian.
In Section 6 we present an algorithm for enumerating small connected quandles that issimilar to but several orders of magnitude faster than the recent algorithm of Vendramin[45]. The outcome of our search agrees with Vendramin’s findings. Using combinatorial andgeometric methods, we construct several families of connected quandles. Thanks to Theorem4.3, the proof of connectedness is often a very simple exercise about conjugation.
In Section 7 we investigate quandles of size p, p2 and 2p, where p is a prime. UsingTheorem 4.3, we first show that if Q is a connected quandle of prime power order thenRMlt(Q) is solvable (Proposition 7.2). We then give two conceptually simple proofs of thefact that every connected quandle of order p is affine. This has been known since [11] and,like in [11], our proof relies on a deep result of Kazarin about conjugacy classes of primepower order. We mention the result of [16] that every connected quandle of order p2 is affine.
In the course of writing this paper, McCarron [33] used Cayley-like representations ofquandles to show that there are no connected quandles of order 2p. We give a new andshorter proof of this fact. First, in Section 7 we use Theorem 4.3 to show that McCarron’sresult follows from a certain theorem about transitive groups of degree 2p (Theorem 8.1).Then we prove Theorem 8.1 in Section 8.
1.5. Notation. We apply all mappings to the right of their arguments, written as a super-script. Thus xα means α evaluated at x. To save parentheses, we use xαβ to mean (xα)β,
while xαβ
stands for x(αβ).
For a given group G and y ∈ G we denote by φy the conjugation map by y, that is,xφy = y−1xy for all x ∈ G. As usual, we often use the shorthand xy instead of xφy , and welet [x, y] = x−1xy. Since (x−1)y = (xy)−1, we denote both of these elements by x−y.
For α ∈ Aut(G) we let CG(α) = {z ∈ G : zα = z} be the centralizer of α, and we writeCG(x) for CG(φx).
If G acts on X and x ∈ X, we let Gx = {g ∈ G : xg = x} be the stabilizer of x, andxG = {xg : g ∈ G} the orbit of x.
3
Note that for any binary algebra (Q, ·), every b ∈ Q and α ∈ Aut(Q) the mapping Rαb is
equal to Rbα , because for every a ∈ Q we have
aRαb = aα
−1Rbα = (aα−1 · b)α = a · bα = aRbα .
Consequently, if Rb is a permutation, then R−αb = (Rαb )−1 = R−1bα . We will use this observa-
tion freely.When A, B are isomorphic algebras, we denote the situation by A ' B.
2. The group of displacements
In this section we present basic properties of quandles and racks in which a certain sub-group of the right multiplication group plays an important role. Most of the material canbe found in Joyce’s papers [22, 23] or even earlier [24, 35].
For a rack Q, define the group of displacements as
Dis(Q) = 〈RaR−1b : a, b ∈ Q〉.
Note that
Dis(Q) ≤ RMlt(Q) ≤ Aut(Q).
For every a, b ∈ Q, we have R−1a Rb ∈ Dis(Q), too, as R−1a Rb = RbR−Rba = RbR
−1aRb
= RbR−1ab .
Proposition 2.1. Let Q be a rack. Then:
(i) RMlt(Q)′ EDis(Q)E RMlt(Q) and the group RMlt(Q)/Dis(Q) is cyclic.(ii) Dis(Q) = {Rk1
a1. . . Rkn
an : n ≥ 0, ai ∈ Q and∑n
i=1 ki = 0}.(iii) If Q is a quandle, the natural actions of RMlt(Q) and Dis(Q) on Q have the same
orbits.
Proof. (i) Let G = RMlt(Q) and D = Dis(Q). For a, b ∈ Q and α ∈ G we have (RaR−1b )α =
Rαa R−αb = RaαR
−1bα ∈ D, proving D E G. Fix e ∈ Q and note that DRa = DRe for every
a ∈ Q. Each element α ∈ G is of the form α = Rk1a1. . . Rkn
an . Then Dα = DRk1+···+kne , proving
G/D = 〈DRe〉. Since G/D is abelian, we obtain G′ ≤ D.(ii) Let S be the set in question. Since the defining generators of D belong to S, and since
S is easily seen to be a subgroup of G, we have D ≤ S. For the other inclusion, we notethat every α ∈ S can be written as Rk1
a1. . . Rkn
an , where not only∑
i ki = 0 but also ki = ±1.Assuming such a decomposition, we prove by induction on n that α ∈ D.
If n = 0 then α = 1, the case n = 1 does not occur, and if n = 2, we have either α = RaR−1b
or α = R−1a Rb, both in D. Suppose that n > 2.If k1 = kn then there is 1 < m < n such that
∑i<m ki = 0 and
∑i≥m ki = 0. Let
β = Rk1a1. . . Rkm−1
am−1and γ = Rkm
am . . . Rknan . Then β, γ ∈ D, and so α = βγ ∈ D.
If k1 6= kn then α = RkaβR
−kb for some a, b ∈ Q, k = ±1 and β = Rk2
a2. . . Rkn−1
an−1. Note
that∑
2≤i≤n−1 ki = 0, hence β ∈ D. We have α = β(Rka)βR−kb = βRk
aβR−kb , and since
RkaβR−kb ∈ D, we are done.
(iii) Let α = Rk1a1. . . Rkn
an ∈ G, and let x, y ∈ Q be such that xα = y. With k = k1+· · ·+kn,
we have β = αR−ky ∈ D by (ii), and xβ = xαR−ky = yR
−ky = y, using idempotence in the last
step. �4
The orbits of transitivity of the group RMlt(Q) (or, equivalently, of the group Dis(Q)) inits natural action on Q will be referred to simply as the orbits of Q. We denote by eQ theorbit containing e. Orbits are subquandles, not necessarily connected.
Example 2.2. In general, proper inclusion RMlt(Q)′ < Dis(Q) can occur. The smallestexample has three elements and two orbits, and is defined by the following Cayley table:
Q 1 2 31 1 1 12 3 2 23 2 3 3
However, in connected racks, the equality RMlt(Q)′ = Dis(Q) always holds:
Proposition 2.3. If Q is a connected rack then RMlt(Q)′ = Dis(Q).
Proof. In view of Proposition 2.1, it remains to prove that the generators RaR−1b of Dis(Q)
belong to RMlt(Q)′. Let α ∈ RMlt(Q) be such that b = aα. Then RaR−1b = RaR
−1aα =
RaR−αa = [R−1a , α] ∈ RMlt(Q)′. �
In particular, we will often use the fact that if Q is a connected quandle, then RMlt(Q)′ =Dis(Q) acts transitively on Q, by Proposition 2.1.
In some cases, the structure of Dis(Q) corresponds to algebraic properties of Q, as thefollowing result illustrates:
Proposition 2.4. Let Q be a rack. Then
(i) Dis(Q) = 1 iff the operation on Q does not depend on the second coordinate. Inquandles, this is equivalent to the operation being the left projection.
(ii) Dis(Q) is abelian iff Q is medial.
Proof. (i) By definition, Dis(Q) = 1 iff Ra = Rb for every a, b ∈ Q. If Q is a quandle, wethen get ab = aRb = aRa = a.
(ii) Note that the following identities are equivalent: Q is medial, RyRuv = RuRyv,RyR
−1v RuRv = RuR
−1v RyRv,
(2.1) RyR−1v Ru = RuR
−1v Ry.
Suppose that Dis(Q) is commutative. Then (RyR−1v )(RuR
−1y ) = (RuR
−1y )(RyR
−1v ) =
RuR−1v , which yields (2.1) upon applying Ry to both sides. Hence Q is medial.
Conversely, if Q is medial, then (2.1) holds, and its inverse yields R−1y RvR−1u = R−1u RvR
−1y ,
so RxR−1y RvR
−1u = RxR
−1u RvR
−1y = RvR
−1u RxR
−1y , where we have again used (2.1) in the
last equality. Hence Dis(Q) is commutative. �
A prototypical example of medial quandles is the following:
Example 2.5. Let (A,+) be an abelian group and f ∈ Aut(A). Define the affine (orAlexander) quandle over the group A as
Aff(A, f) = (A, ∗), x ∗ y = xf + y1−f .
Straightforward calculation shows that (A, ∗) is a quandle. For mediality, observe that
Alternatively, given an R-module M and an invertible element r ∈ R, then (M, ∗) with
x ∗ y = xr + y(1− r)is an affine quandle over the group (M,+). The two definitions are equivalent. (Withoutloss of generality, we can assume R = Z[t, t−1], the ring of Laurent series.)
Affine quandles are not necessarily connected, and most medial quandles are not affine.(The smallest non-affine medial quandle is the one in Example 2.2.) However, we prove laterthat all connected medial quandles are affine.
3. Galkin representations and the minimal representation
In this section and the next one we present two representations of connected quandlesbased on transitive permutation groups: the minimal representation of Theorem 3.5 and thecanonical representation of Theorem 4.3. Most of our work here is inspired by Galkin [12],who discovered analogous representations for latin quandles.
The starting point is the following well-known construction, which generalizes the affinequandles from Example 2.5:
Construction 3.1. Let G be a group, H ≤ G, f ∈ Aut(G), and suppose that H ≤ CG(f).Denote by G/H the set of right cosets {Hx : x ∈ G}. Define
(3.1) Gal(G,H, f) = (G/H, ∗), Hx ∗Hy = H(xy−1)fy.
First we note that the operation ∗ is well defined. Indeed, if Hx = Hu and Hy = Hvthen u = hx, v = ky for some h, k ∈ H, and
H(uv−1)fv = H(hxy−1k−1)fky = Hhf (xy−1)f (k−1)fky
= Hh(xy−1)fk−1ky = H(xy−1)fy,
using H ≤ CG(f).In fact, Gal(G,H, f) is always a quandle. Idempotence is immediate from
where in the last step we applied f−1 to both sides and used H ≤ CG(f). Hence, given Hy,Hz, the equation Hx ∗Hy = Hz has a unique solution Hx.
We say that a quandle is Galkin representable if it is isomorphic to a quandle Gal(G,H, f)from Construction 3.1.
6
Example 3.2. Affine quandles are Galkin representable. Indeed, let (A,+) be an abeliangroup and f ∈ Aut(A). Then Aff(A, f) = Gal(A, 0, f), with x∗y = (x−y)f +y = xf +y1−f .
Example 3.3. Knot quandles are Galkin representable. Let K be a knot, and GK = π1(UK)the knot group where UK is the complement of a tubular neighborhood of K. Let HK bethe peripheral subgroup of GK and fK conjugation by the meridian. Then Gal(GK , HK , fK)is the knot quandle of K. See [22, Corollary 16.2] or [32, Proposition 2] for details.
Not every quandle is Galkin representable, for instance, the one in Example 2.2 is not.However, every connected quandle and, more generally, every quandle orbit is Galkin repre-sentable. Before we prove this in Theorem 3.5, we need an auxiliary result.
In the special case of Gal(G,H, f) when G is a permutation group on a set Q and H = Ge
for some e ∈ Q, we define the mapping
(3.2) πe : Gal(G,Ge, f)→ eG, Geα 7→ eα.
Since Geα = Geβ holds if and only if eα = eβ, the mapping πe is well-defined and bijective.
Proposition 3.4. Let Q be a quandle and e ∈ Q. Let G be the right multiplication groupRMlt(Q) or the displacement group Dis(Q). Let f be the restriction of the conjugation byRe in RMlt(Q) onto G. Then Gal(G,Ge, f) is well defined and isomorphic to the orbit eQ.
Proof. Since f is a restriction of the conjugation by Re ∈ RMlt(Q) onto a normal subgroupG of RMlt(Q), it is indeed an automorphism of G. To check Ge ≤ CG(f), consider α ∈ Ge.For every x ∈ Q we have xαRe = xα · e = xα · eα = (xe)α = xReα and so αRe = α as required.The quandle Gal(G,Ge, f) is therefore well defined, with multiplication
Geα ∗Geβ = Ge(αβ−1)fβ = GeR
−1e αβ−1Reβ.
Consider the bijective mapping πe from (3.2). By Proposition 2.1(iii), eG = eQ. To see thatπe is a homomorphism, we calculate
(Geα ∗Geβ)πe = eR−1e αβ−1Reβ = (eαβ
−1 · e)β = eα · eβ = (Geα)πe · (Geβ)πe ,
where we have used eRe = e and β ∈ Aut(Q). �
Given a connected quandle Q and an element e ∈ Q, we will call the Galkin representationQ ' Gal(G,Ge, φRe) of Proposition 3.4
• the canonical representation of Q if G = RMlt(Q),• the minimal representation of Q if G = RMlt(Q)′ = Dis(Q).
We will discuss canonical representations in the next section. The following result explainswhy we have used the adjective “minimal.”
Theorem 3.5 (Minimal representation of connected quandles). Let Q be a connected quan-dle. Then:
(i) Q ' Gal(G,H, f) whenever G = RMlt(Q)′, e ∈ Q, H = Ge and f = φRe is theconjugation by Re on G.
(ii) If Q ' Gal(G,H, f) for some G, H ≤ G and f ∈ Aut(G), then RMlt(Q)′ embedsinto a quotient of G.
7
Proof. Part (i) is just Proposition 3.4 with Q = eQ. Let us therefore assume the hypothesisof part (ii), where for simplicity we take Q = Gal(G,H, f). Define ϕ : G → Aut(Q) bya 7→ ϕa, where (Hx)ϕa = Hxa. The mappings ϕa are automorphisms of Q, since
The mapping ϕ is obviously a homomorphism. We show that RMlt(Q)′ is a subgroup ofIm(ϕ), and hence that RMlt(Q)′ embeds into G/Ker(ϕ).
By Proposition 2.3, RMlt(Q)′ = Dis(Q). It therefore suffices to check that RHxR−1Hy ∈
Im(ϕ) for every x, y ∈ G. Recall that the unique solution to Hx ∗ Hy = Hz is Hx =
H(zy−1)f−1y, and thus (Hz)R
−1Hy = H(zy−1)f
−1y. Hence for every x, y, u ∈ G we have
(Hu)RHxR−1Hy = (H(ux−1)fx)R
−1Hy = H((ux−1)fxy−1)f
−1
y = Hux−1(xy−1)f−1
y,
proving RHxR−1Hy = ϕx−1(xy−1)f−1y. �
Corollary 3.6. Let Q be a finite connected quandle, and let G be of smallest order amongall groups such that Q ' Gal(G,H, f). Then G ' RMlt(Q)′.
4. The canonical representation
Throughout this section, fix a set Q and an element e ∈ Q. We proceed to establish a one-to-one correspondence between connected quandles defined on Q and certain configurationsin transitive groups on Q that we will call quandle envelopes.
A quandle folder is a pair (G, ζ) such that G is a transitive group on Q and ζ ∈ Z(Ge).A quandle folder (G, ζ) is a quandle envelope if also 〈ζG〉 = G, that is, if the smallest normalsubgroup of G containing ζ is all of G.
For a connected quandle (Q, ·), define
E(Q) = (RMlt(Q), Re).
Lemma 4.1. Let Q be a connected quandle and e ∈ Q. Then E(Q) is a quandle envelope.
Proof. Let G = RMlt(Q). Note that Re ∈ Ge. With α ∈ Ge ≤ Aut(Q), we calculatexαRe = xα · e = xα · eα = (xe)α = xReα, so Re ∈ Z(Ge). Since Q is connected, G actstransitively on Q, and for every x ∈ Q there is x ∈ G such that ex = x. Then R x
e = Rex = Rx,proving 〈RG
e 〉 = G. �
For a quandle folder (G, ζ), define
Q(G, ζ) = (Q, ◦), x ◦ y = xζy
,
where y is any element of G satisfying ey = y. We shall see that the operation does notdepend on the choice of the permutations y, and that Q(G, ζ) is Galkin representable.
Lemma 4.2. Let (G, ζ) be a quandle folder on the set Q with a fixed element e ∈ Q. Then:
(i) If α, β ∈ G satisfy eα = eβ then ζα = ζβ.(ii) The definition of Q(G, ζ) does not depend on the choice of the permutations y.
(iii) The mapping πe of (3.2) is an isomorphism of Gal(G,Ge, φζ) onto Q(G, ζ).(iv) Q(G, ζ) is a quandle.(v) RMlt(Q(G, ζ)) = 〈ζ y : y ∈ Q〉 = 〈ζG〉.
(vi) If (G, ζ) is a quandle envelope, then Q(G, ζ) is a connected quandle.8
Proof. For α, β ∈ G, note that ζα = ζβ iff β−1α commutes with ζ. The latter conditioncertainly holds when eα = eβ because ζ ∈ Z(Ge). This proves (i), and part (ii) follows.
Consider again the bijection πe of (3.2). SinceG is transitive, πe is ontoQ. To check that πe
is a homomorphism, note that ζβ = ζ eβ
by (i). Therefore, with Gal(G,Ge, φζ) = (G/Ge, ∗),we have Geα ∗Geβ = Ge(αβ
−1)ζβ = Geζ−1αζβ = Geαζ
β, and thus
(Geα ∗Geβ)πe = (Geαζβ)πe = eαζ
β
= (eα)ζeβ
= eα ◦ eβ = (Geα)πe ◦ (Geβ)πe .
This proves (iii), and part (iv) follows.For (v), note that the right translation by y in (Q, ◦) is the mapping ζ y and, once again,
ζβ = ζ eβ
for any β ∈ G. Part (vi) follows. �
Theorem 4.3 (Canonical representation of connected quandles). Let Q be a set and e ∈ Q.Then the mappings
E : Q 7→ (RMlt(Q), Re),
Q : (G, ζ) 7→ (Q, ◦), x ◦ y = xζy
are mutually inverse bijections between the set of connected quandles and the set of quandleenvelopes on the set Q.
Proof. In view of Lemmas 4.1 and 4.2, it remains to show that the two mappings are mutuallyinverse. Let (G, ζ) be a quandle envelope, and let (Q, ◦) = Q(G, ζ) be the correspondingconnected quandle. Then RMlt(Q, ◦) = 〈ζG〉 = G by Lemma 4.2. Moreover, xRe = x ◦ e =
xζe
= xζ thanks to e ∈ Ge and ζ ∈ Z(Ge), hence ζ is the right translation by e in (Q, ◦). Itfollows that E(Q(G, ζ)) = (G, ζ).
Conversely, let Q be a connected quandle, and E(Q) = (RMlt(Q), Re) the corresponding
quandle envelope. Then, in Q(E(Q)), we calculate x ◦ y = xRye = xRy = xy. It follows that
Q = Q(E(Q)). �
Example 4.4. Let K be a knot, GK its knot group, and QK its knot quandle. Then GK actstransitively on QK , and the stabilizer of a fixed element e ∈ Q is the peripheral subgroupHK . Since HK ' Z × Z, the meridian mK is central in the stabilizer, and it follows fromWirtinger’s presentation of GK that GK = 〈mGK
K 〉. We proved that (GK ,mK) is a quandleenvelope, and the knot quandle is isomorphic to Q(GK ,mK). See [22, Section 16] or [32,Section 6] for details.
We conclude this section by solving the isomorphism problem for canonical representa-tions. We will take advantage of this result in Algorithm 6.1, where we enumerate allconnected quandles of given size up to isomorphism. We start with a useful characterizationof isomorphisms between connected quandles in canonical representation.
Lemma 4.5. Let (G, ζ), (K, ξ) be quandle envelopes on a set Q with a fixed element e ∈ Q,and let
(i) A be the set of all quandle isomorphisms ϕ : Q(G, ζ)→ Q(K, ξ) such that eϕ = e;(ii) B be the set of all permutations ϕ of Q such that eϕ = e, ζϕ = ξ and Gϕ = K;
(iii) C be the set of all group isomorphisms ψ : G→ K such that ζψ = ξ and Gψe = Ke.
Then A = B and ϕ 7→ φϕ is a bijection from A = B to C.
9
Proof. Let f denote the mapping ϕ 7→ φϕ defined on B. We show that A ⊆ B, that f mapsB into C, and construct a mapping g : C → A ⊆ B such that fg is the identity mapping onB and gf is the identity mapping on C.
Let Q(G, ζ) = (Q, ◦), where x ◦ y = xζy
for some y ∈ G satisfying ey = y, and Q(K, ξ) =
(Q, ∗), where x∗ y = xξy
for some y ∈ K such that ey = y. Note that the following identitiesare equivalent for a permutation ϕ of Q:
(x ◦ y)ϕ = (xϕ) ∗ (yϕ), (xζy
)ϕ = (xϕ)ξyϕ
, xϕ−1ζyϕ = xξ
yϕ
.
Hence ϕ is an isomorphism (Q, ◦)→ (Q, ∗) if and only if
(ζ y)ϕ = ξyϕ.
We will use this fact freely, as well as Lemma 4.2.(A ⊆ B): We need to show ζϕ = ξ and Gϕ = K. Since eϕ = e, we have ζϕ = (ζ e)ϕ =
ξeϕ
= ξe = ξ. To prove Gϕ ⊆ K, note that G = 〈ζG〉, pick α ∈ G, and calculate (ζα)ϕ =
(ζ eα)ϕ = ξe
αϕ ∈ K. For the other inclusion K ⊆ Gϕ, note that K = 〈ξK〉, pick β ∈ K, find
α ∈ G such that eβ = eαϕ by transitivity of G, and calculate ξβ = ξeβ
= ξeαϕ
= (ζ eα)ϕ ∈ Gϕ.
(f : B → C): For ϕ ∈ B let ψ = ϕf = φϕ be the conjugation by ϕ. Since Gϕ = K, we seethat ψ is an isomorphism G→ K. Clearly ζψ = ζϕ = ξ. To verify Gψ
e = Ke, let α ∈ Ge and
calculate eαψ
= eαϕ
= eϕ−1αϕ = e, so αψ ∈ Ke.
(g : C → A): For ψ ∈ C, define ϕ = ψg by
xϕ = exψ
for every x ∈ Q. We show that ϕ is an isomorphism (Q, ◦)→ (Q, ∗) that fixes e. The second
condition follows immediately from eϕ = eeψ
= e, because e ∈ Ge and Gψe = Ke.
Let us observe two facts. First, if α, β ∈ G, then the following conditions are equivalent:
eαψ
= eβψ
, eβψ(αψ)−1
= e, (βα−1)ψ ∈ Ke, βα−1 ∈ Ge, eα = eβ.
This implies that ϕ is a bijection. Second, for any x ∈ Q and α ∈ G we have exα
= xα = exα.Combining the two observations, we see that
(4.1) exαψ
= e(xα)ψ
.
For x, y ∈ Q, we then have
(x ◦ y)ϕ = ex◦yψ
= exζyψ
= e(xζy)ψ = ex
ψ(ζy)ψ
= (xϕ)(ζy)ψ = (xϕ)(ζ
ψ)(yψ)
= (xϕ)ξ(y ψ)
= (xϕ)ξyϕ
= xϕ ∗ yϕ,
where in the penultimate step we used eyψ
= yϕ.(fg = id): For ϕ ∈ B and x ∈ Q we have
xϕfg
= x(ϕf )g = ex
(ϕf )
= exϕ
= eϕ−1xϕ = exϕ = xϕ.
(gf = id): For ψ ∈ C and α ∈ G, we would like to show that αψgf
= α(ψg)f = αψg
is equal
to αψ. With x ∈ Q, and keeping (4.1) in mind, set u = x(ψg)−1
for brevity, and calculate
xαψg
= x(ψg)−1αψg = (uα)ψ
g
= euαψ
= e(uα)ψ
= euψαψ = (uψ
g
)αψ
= xαψ
.
�10
A solution to the isomorphism problem now easily follows:
Theorem 4.6. Let (G, ζ), (K, ξ) be quandle envelopes on a set Q with a fixed element e ∈ Q.Then the following conditions are equivalent:
(i) Q(G, ζ) ' Q(K, ξ).(ii) There is a permutation ϕ of Q such that eϕ = e, ζϕ = ξ and Gϕ = K.
(iii) There is an isomorphism ψ : G→ K such that ζψ = ξ and Gψe = Ke.
Proof. Let ρ : Q(G, ζ) → Q(K, ξ) be an isomorphism, and let α ∈ K be such that eρα = e.Since α ∈ K = RMlt(Q(K, ξ)) ≤ Aut(Q(K, ξ)) by Theorem 4.3, the permutation ϕ = ραis also an isomorphism Q(G, ζ) → Q(K, ξ) and it satisfies eϕ = e. The rest follows fromLemma 4.5. �
In particular, isomorphic connected quandles have isomorphic right multiplication groups,and their right translations have the same cycle structure.
A given transitive group can represent many connected quandles, depending on the choiceof ζ. Upon specializing Theorem 4.6 to the case G = K, we obtain:
Corollary 4.7. Let (G, ζ), (G, ξ) be quandle envelopes on a set Q with a fixed element e ∈ Q.Then Q(G, ζ) is isomorphic to Q(G, ξ) if and only if ζ and ξ are conjugate in N(SQ)e(G),the normalizer of G in the stabilizer of e in the symmetric group SQ.
Another application of Lemma 4.5 reveals the structure of the automorphism group of aconnected quandle in terms of its right multiplication group. For a group G, a subgroupH ≤ G and an element x ∈ G we let
Proposition 4.8. Let Q be a connected quandle, e ∈ Q, and let G = RMlt(Q). ThenAut(Q) is isomorphic to (Go Aut(G)Re,Ge) /{(α, φ−1α ) : α ∈ Ge}.
Proof. By Theorem 4.3, we have Q = Q(G,Re). According to Lemma 4.5, ϕ 7→ φϕ is abijection between Aut(Q)e and Aut(G)Re,Ge , which is easily seen to be a homomorphism.Define f : Go Aut(Q)e → Aut(Q) by (α, ϕ)f = αϕ. This is a homomorphism, since
(α, ϕ)f (β, ψ)f = αϕβψ = αβϕ−1
ϕψ = ((α, ϕ)(β, ψ))f .
Since G acts transitively on Q, every ψ ∈ Aut(Q) can be decomposed as ψ = αϕ, whereα ∈ G and ϕ ∈ Aut(Q)e. Thus f is surjective. The kernel of f consists of all tuples (α, ϕ)with αϕ = 1, hence ϕ = α−1 ∈ G ∩ Aut(Q)e = Ge. �
5. Connected affine quandles
Let A = (A,+) be an abelian group. The set
Aff(A) = {x 7→ c+ xf : c ∈ A, f ∈ Aut(A)}is a subgroup of the symmetric group SA, and the elements of Aff(A) are called affinemappings over A. (Note that Aff(A) is isomorphic to A o Aut(A), the holomorph of A,where the mapping x 7→ c+ xf corresponds to the pair (c, f) ∈ Ao Aut(A).) The set
Mlt(A) = {x 7→ c+ x : c ∈ A}is a subgroup of Aff(A), and its elements are called translations.
11
Recall that a quandle (A, ∗) = Aff(A, f) is called affine if there is an abelian group (A,+)and an automorphism f ∈ Aut(A) such that x ∗ y = xf + y1−f . Thus, in (A, ∗),
xRy = xf + y1−f ,
xR−1y = xf
−1
+ y1−f−1
,
hence the right translations are affine mappings over A and RMlt(Aff(A, f)) ≤ Aff(A).The following characterization of connected affine quandles is well known. Note that the
equality (−f−1)(1− f) = 1− f−1 implies that Im(1− f) = Im(1− f−1), which we will useon two occasions.
Proposition 5.1. An affine quandle Aff(A, f) is connected if and only if 1− f is onto.
Proof. Let Q = Aff(A, f), G = RMlt(Q), and let 0 be the identity element of (A,+). Itsuffices to prove that the orbit 0G is equal to Im(1−f). If x ∈ Im(1−f) then x = y1−f = 0Ry
for some y ∈ A, proving Im(1− f) ⊆ 0G.For the converse, we note that 0 ∈ Im(1− f), and we prove that whenever x ∈ Im(1− f)
then also xRy , xR−1y ∈ Im(1 − f). If x = u1−f ∈ Im(1 − f) then xRy = u(1−f)f + y1−f =
(uf + y)1−f ∈ Im(1 − f), and xR−1y = u(1−f)f
−1+ y1−f
−1= (y − u)1−f
−1 ∈ Im(1 − f−1) =Im(1− f). �
Remark 5.2. It is easy to check that affine quandles are both left and right distributive,that is, satisfy x ∗ (y ∗ z) = (x ∗ y) ∗ (x ∗ z) in addition to (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z). Itfollows from Proposition 5.1 that a finite affine quandle is connected if and only if it is latin.A stronger result is proved in [5, Theorem 5.10]: A finite distributive quandle is connected ifand only if it is latin.
Infinite connected affine quandles need not be latin, however. Indeed, in Aff(Zp∞ , 1− p),the mapping 1− (1− p) = p is onto but not one-to-one.
We are now going to prove a somewhat surprising result that a connected quandle is affineif and only if it is medial. (Recall that there are medial quandles that are not affine, asillustrated by Example 2.2.)
In the finite case, the result follows from [5, Theorem 5.10] mentioned above, and fromthe Toyoda-Bruck theorem [4] that states that medial quasigroups are affine. Our methodis substantially different, includes the infinite case, and provides a new proof of a specialcase of the Toyoda-Bruck theorem for idempotent medial quasigroups (it does not extend tonon-idempotent quasigroups in a straightforward fashion).
The crucial point in Theorem 5.3 is characterization (iv), which is interesting on its ownand will be used in Section 7.
Theorem 5.3. The following conditions are equivalent for a connected quandle Q:
(i) Q is affine.(ii) Q is medial.
(iii) RMlt(Q)′ is abelian.(iv) There is an abelian group A = (Q,+) such that Mlt(A) ≤ RMlt(Q) ≤ Aff(A).
Proof. (i) ⇒ (ii) ⇒ (iii): We have already seen in Example 2.5 that every affine quandle ismedial. By Propositions 2.3 and 2.4, every connected medial quandle Q has RMlt(Q)′ =Dis(Q) abelian.
12
(iii)⇒ (iv): Fix e ∈ Q. Since RMlt(Q)′ is abelian and transitive (by Propositions 2.1 and2.3), it is sharply transitive. Thus for every y ∈ Q there is a unique y ∈ RMlt(Q)′ such thatey = y. Define A = (Q,+) by
x+ y = xy.
We claim that ϕ : A→ RMlt(Q)′, x 7→ x is an isomorphism and hence that A is an abelian
group. Indeed, ϕ is clearly a bijection, we have exy
= xy = exy, thus xy = xy by sharp
transitivity, and then (x+ y)ϕ = (xy)ϕ = xy = xy = xϕyϕ.Since the right translation by y in A is y ∈ RMlt(Q)′, we have Mlt(A) = RMlt(Q)′ ≤
RMlt(Q). To prove that RMlt(Q) ≤ Aff(A), it suffices to show that Re ∈ Aut(A) andx · y = xRe + y1−Re , because then Ry ∈ Aff(A) for every y ∈ Q. We have Re ∈ Aut(A)
iff (x + y)Re = xyRe is equal to xRe + yRe = xReyRe for every x, y ∈ Q, which is equivalent
to yRe = yRe for every y ∈ Q. Taking advantage of sharp transitivity, the last equality is
verified by eyRe
= y · e = eyRe . We have Q = Q(E(Q)) by Theorem 4.3, and hence
x · y = xRye = xy
−1Rey = (x− y)Re + y = y1−Re + xRe .
(iv) ⇒ (i): Let 0 be the identity element of A = (Q,+). Fix y ∈ Q and denote by ρy the
right translation by y in A. By Theorem 4.3, we have Ry = R y0 for some y ∈ RMlt(Q) such
that 0y = y. Since RMlt(Q) ≤ Aff(A), there are c ∈ Q and g ∈ Aut(A) such that xy = c+xg
for every x ∈ Q. But c = 0y = y, so xy = y + xg and y = gρy. Since Mlt(A) ≤ RMlt(Q),we have g = yρ−1y ∈ RMlt(Q). Hence g ∈ RMlt(Q)0, and since R0 ∈ Z(RMlt(Q)0), we
obtain gR0 = R0g. Since 0R0 = 0 by idempotence, we have not only R0 ∈ Aff(A) but in factR0 ∈ Aut(A). Using all these facts, we calculate
x · y = xRy0 = xy
−1R0y = xρ−1y g−1R0gρy = xρ
−1y R0ρy = (x− y)R0 + y = xR0 + y1−R0
for every x, y ∈ Q, proving that Q = Aff(A,R0) is an affine quandle. �
We finish this section with a brief discussion of the isomorphism problem and enumerationof connected affine quandles. Most of the ideas appeared in some form earlier [1, 19].
Proposition 5.4. Let Q = Aff(A, f) be an affine quandle. Then Dis(Q) ' Im(1− f).
Proof. We show that Dis(Q) is equal to T = {z 7→ z+c : c ∈ Im(1−f)}. Then the mappingϕ : Im(1− f)→ Dis(Q) which maps c to the translation by c is an isomorphism. Note thatT is a group.
(Dis(Q) ⊆ T ): We calculate
zRxR−1y = (zf + x1−f )f
−1
+ y1−f−1
= z + x(1−f)f−1
+ y1−f−1
,
so zRxR−1y = z + c with the constant c = x(1−f)f
−1+ y1−f
−1 ∈ Im(1 − f) + Im(1 − f−1) =Im(1− f). The defining generators of Dis(Q) are therefore in T , and Dis(Q) ≤ T follows.
(Dis(Q) ⊇ T ): Given c ∈ Im(1 − f), choose x ∈ A so that x(1−f)f−1
= c, and verify that
zRxR−10 = (zf + x1−f )f
−1= z + c. �
Corollary 5.5. Let (A, ∗) = Aff(A, f) be a connected affine quandle. Then the isomorphismtype of the abelian group (A,+) can be recovered from (A, ∗) without any knowledge of f .
Proof. Propositions 5.1 and 5.4 imply that A = Im(1− f) ' Dis(A, ∗). �13
In particular, if A, B are abelian groups such that A 6' B, then Aff(A, f) 6' Aff(B, g) forany f ∈ Aut(A), g ∈ Aut(B) with 1− f and 1− g onto. In general, an affine quandle thatis not connected can sometimes be constructed from several non-isomorphic abelian groups.This phenomenon will be discussed in detail in [18].
Remark 5.6. Murillo et al. [34] asked how to determine whether a quandle, given by itsCayley table, is affine, and when it is, how to find its affine representation. They provideda simple but inefficient algorithm based on a couple of necessary conditions and a bruteforce search. We note that for a connected quandle Q the problem is rather easy, thanks toTheorem 5.3, as it suffices to test whether G = RMlt(Q)′ is abelian, and in the positive casereturn A = G and f = φRe for any e ∈ Q. An efficient test of affinity for general quandleswill be presented in [18].
Proposition 5.7. Let Aff(A, f), Aff(A, g) be connected affine quandles. Then Aff(A, f) isisomorphic to Aff(A, g) if and only if f and g are conjugate in Aut(A).
Proof. Suppose that g = fϕ for some ϕ ∈ Aut(A). Then
(xf + y1−f )ϕ = xϕfϕ
+ yϕ − yϕfϕ = (xϕ)g + (yϕ)1−g
shows that ϕ is an isomorphism Aff(A, f)→ Aff(A, g).Conversely, let ϕ be an isomorphism Aff(A, f) → Aff(A, g). Then (xf + y1−f )ϕ = xϕg +
yϕ(1−g) for every x, y ∈ A, and taking y = 0 yields xfϕ = xϕg, that is, g = fϕ. Since 1− f isonto A by Proposition 5.1, given x, y ∈ A there are u, v ∈ A such that x = uf and y = v1−f .Then
Corollary 5.5 and Proposition 5.7 can be used to enumerate finite connected affine quandlesup to isomorphism. It suffices to consider abelian groups of a given order up to isomorphism,and for each such group A to determine all f ∈ Aut(A) with 1− f also in Aut(A), where itsuffices to consider f up to conjugation in Aut(A).
For example, for a prime order p, we can assume A = Zp and consider all f ∈ Aut(A) ' Z∗psuch that 1 − f 6= 0, that is, f 6= 1. Since Aut(A) is abelian, conjugacy plays no role, andwe obtain p− 2 connected affine quandles with p elements.
In [19], Hou proved a stronger result than Proposition 5.7, solving the isomorphism prob-lem for all finite affine quandles (not necessarily connected). Using the method describedabove, he found explicit formulas for the number of affine quandles up to isomorphism withpk elements, k = 1, 2, 3, 4, both in the general and the connected cases. For example, on p2
elements, there are precisely 2p2 − 3p − 1 connected affine quandles, p2 − 2p with A = Zp2and p2 − p− 1 with A = Zp × Zp. (According to Theorems 7.3 and 7.4, all quandles with pand p2 elements are affine.)
We also note that a variant of Proposition 5.7 in the setting of distributive quasigroupswas proved by Kepka and Nemec [26]. They used it to show that non-medial distributivequasigroups exist only on 3k elements, k ≥ 4, and enumerated them for k = 4, 5. (In thequandle terminology, we speak of latin distributive quandles. Recall that all finite connecteddistributive quandles are latin [5].)
14
6. Enumerating small connected quandles
Suppose that we wish to enumerate all connected quandles of order n up to isomorphism.By Theorem 4.3, it suffices to fix a set Q of size n and an element e ∈ Q, and consider allquandle envelopes (G, ζ), where G is a transitive group on Q, and where ζ ∈ Z(Ge) satisfies〈ζG〉 = G. The corresponding connected quandle Q is then (Q, ◦) = Q(G, ζ).
Moreover, since E(Q(G, ζ)) = (G, ζ) by Theorem 4.3, we see that G = RMlt(Q) (andζ = Re). Propositions 2.1 and 2.3 then imply that it suffices to consider transitive groups Gfor which G′ is also transitive and G/G′ is cyclic. This disqualifies many transitive groups.
In principle, Theorem 4.6 then solves the isomorphism problem: Given two quandle en-velopes (G, ζ) and (K, ξ), the connected quandles Q(G, ζ) and Q(K, ξ) are isomorphic if andonly if G is isomorphic to K, i.e., G = K if we start with a list of transitive groups up toisomorphism, and if ζ, ξ are conjugate in the normalizer N(SQ)e(G).
In practice, to check whether ζ, ξ are conjugate in N(SQ)e(G) is costly, and we can use adirect isomorphism check on all quandles constructed from all quandle envelopes (G, ζ) witha fixed transitive group G. Here is the resulting algorithm for a given size n:
Algorithm 6.1.quandles ← ∅for each G in the set of transitive groups on {1, . . . , n} up to isomorphism do
if G′ is transitive and G/G′ is cyclic then
qG ← ∅for each ζ in Z(G1) such that 〈ζG〉 = G do
qG ← qG ∪ {Q(G, ζ)}end
qG ← qG filtered up to isomorphism
quandles ← quandles ∪ qG
end
end
return quandles
We have implemented the algorithm in GAP [14], and the source code and the outputof the search are available on the website of the third author. The isomorphism check isbased on the methods of the LOOPS package for GAP. The current version of GAP containsa library of transitive groups up to degree 30, and an extension up to degree 35 can beobtained from its authors [20].
The power of Theorem 4.3 is tremendous. On an Intel Core i5-2520M 2.5GHz processor,the search for all connected quandles of order 1 ≤ n ≤ 35 with n 6= 32 takes only about 5minutes, and the order n = 32 takes about an hour.
A similar algorithm was presented by Vendramin [45]. He was not aware of Theorem 4.3,and his algorithm is based on a weaker representation, analogous to our Proposition 3.4with G = RMlt(Q). Consequently, he had to deal with many more transitive groups, hadto filter out quandles that are not connected, and also had to filter many quandles up toisomorphism, resulting in a much longer computation time (on the order of weeks).
Table 1 shows the number q(n) of connected quandles of size n, the number `(n) oflatin quandles of size n, and the number a(n) of connected affine quandles of size n, up toisomorphism. Latin quandles are recognized by a direct test whether all left translations
Table 1. The numbers q(n) of connected quandles, `(n) of latin quandles,and a(n) of connected affine quandles of size n up to isomorphism.
are permutations. Affine quandles are recognized by checking whether G′ is abelian, usingProposition 2.4 and Theorem 5.3. Note that Proposition 5.1 implies a(n) ≤ `(n) ≤ q(n),while Stein’s theorem [43] gives `(4k + 2) = 0.
The numbers q(n) agree with those calculated by Vendramin in [45], and the numbers a(n)agree with the enumeration results of Hou [19] (see the discussion at the end of Section 5).Note that if m, n are coprime then a(mn) = a(m)a(n), hence Hou’s formulas yield all valuesof a(n) in Table 1 except for a(32).
We conclude this section by providing examples of sequences of connected quandles. Thefirst source of examples is combinatorial, resulting from multitransitivity of the symmetricand alternating groups.
Example 6.2. For n ≥ 2 let G = Sn act on 2-element subsets of {1, . . . , n}, let e = {1, 2}and ζ = (1, 2). Then ζ ∈ Z(Ge) and 〈ζG〉 = G, since all transpositions are conjugate to ζ inSn. Thus Q(G, ζ) is a connected quandle of order
(n2
).
Example 6.3. For n ≥ 2 let G = Sn act on n-cycles by conjugation, let e = (1, . . . , n)and ζ = (1, . . . , n). Since the orbit of e consists of all n-cycles, we see that |Ge| = n andGe = Z(Ge) = 〈ζ〉, so certainly ζ ∈ Z(Ge). Furthermore, 〈ζG〉 generates Sn if n is even (andAn if n is odd). Therefore, if n is even then Q(G, ζ) is a connected quandle of order (n− 1)!.
Example 6.4. For n ≥ 3 let G = Sn act on (n − 2)-tuples of distinct elements pointwise,let e be the (n − 2)-tuple (1, . . . , n − 2), and let ζ = (n − 1, n). Then we obviously haveGe = Z(Ge) = 〈ζ〉, so ζ ∈ Z(Ge), and 〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle oforder n!/2.
Example 6.5. For n ≥ 4 let G = An act on (n − 3)-tuples of distinct elements pointwise,let e be the (n− 3)-tuple (1, . . . , n− 3), and let ζ = (n− 2, n− 1, n). Since |Ge| = 6/2 = 3(because G = An, rather than G = Sn), we have Ge = Z(Ge) = 〈ζ〉, so ζ ∈ Z(Ge). As An isgenerated by 3-cycles, we also have 〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle of ordern!/6.
There are also geometric constructions, as illustrated by the following examples:
Example 6.6. For a prime power q, let G = SL2(q) act (on the right) on Q, the set ofall non-zero vectors in the plane (Fq)2. Let e = (1, 0). A quick calculation shows that
16
Ge = {Ma : a ∈ Fq}, where Ma = ( 1 0a 1 ). Let ζ = M1. Since MaMb = Ma+b, we have
Ge ' (Fq,+), so ζ ∈ Z(Ge) = Ge. We claim that 〈ζG〉 = G.First, it is easy to check that Ma is conjugate to ζ in G if and only if a is a square in Fq. If
q is even then F∗q has odd order q− 1 and thus every element of Fq is a square, so Ge ≤ 〈ζG〉.When q is odd then F∗q contains (q − 1)/2 squares, so there are at least (q − 1)/2 + 1 > q/2
elements in 〈ζG〉 ∩Ge conjugate to ζ, and Lagrange’s Theorem then implies that Ge ≤ 〈ζG〉again.
Since Ge ≤ 〈ζG〉, we establish 〈ζG〉 = G by proving that 〈ζG〉 acts transitively on Q.Given (x, y) ∈ Q with y 6= 0, we have (x, y) = eDM−yD
−1 with D = ( 0 1−1 d ), d = (1− x)y−1.
In particular, (0, 1) ∈ e〈ζG〉, and given (x, 0) ∈ Q, we obtain (x, 0) = (0, 1)EMxE
−1 withE =
(1 x−1
0 1
).
We have proved that 〈ζG〉 = G, and thus Q(G, ζ) is a connected quandle of order q2 − 1.
Example 6.7. For a prime power q, let G = PSL3(q) act on Q, the set of all two-elementsubsets of the projective plane P2(Fq). This is a transitive action, because the natural actionof G on P2(Fq) is 2-transitive. Pick a two-element subset e = {e1, e2} arbitrarily, and considermatrices with respect to the basis (e1, e2, e3), with an arbitrary completion by e3. Clearly,Ge = {Ma,b, Na,b : a, b ∈ Fq}, where
Ma,b =(
1 0 00 1 0a b 1
), Na,b =
(0 1 01 0 0a b −1
).
A quick calculation shows that ζ = Ma,−a ∈ Z(Ge) for every a ∈ Fq. Since G is a simplegroup, we obtain for free that the normal subgroup 〈ζG〉 is equal to G (unless a = 0). ThusQ(G, ζ) is a connected quandle of order |Q| = (q2 + q + 1)(q2 + q)/2.
Example 6.8. The group G of rotations of a Platonic solid (see [7, p.136]) acts on faces.Let e be a face.
• Tetrahedron: We have G = A4 acting on 4 points (faces), and with ζ a generator ofGe ' Z3 we get 〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle of order 4. In fact,since A4 is metabelian, Theorem 5.3 implies that Q(G, ζ) is affine.• Cube: We have G = S4 acting on 6 points, and with ζ a generator of Ge ' Z4 we get〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle of order 6.• Octahedron: We have G = S4 acting on 8 points, and Ge ' Z3. Since 3-cycles do
not generate S4, no choice of ζ ∈ Ge yields a connected quandle Q(G, ζ).• Dodecahedron: We have G = A5 acting on 12 points, and with ζ a generator ofGe ' Z5 we get 〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle of order 12.• Icosahedron: We obtain G = A5 acting on 20 points, and with ζ a generator ofGe ' Z3 we get 〈ζG〉 = G. Thus Q(G, ζ) is a connected quandle of order 20.
On the other hand, there are algebraic constructions where the quandle envelope is notobvious. For example, a general construction of connected quandles of size 3n was presentedby Clark et al. [5], inspired by Galkin [13], by extending the affine quandle Aff(Z3,−1) bya pointed abelian group.
Example 6.9. Let A be an abelian group and c ∈ A. We define µ, τ : Z3 → A by 0µ = 2,1µ = 2µ = −1 and 0τ = 1τ = 0, 2τ = c, and we define a binary operation on Z3 × A by
(x, a) ◦ (y, b) = (−x− y,−a+ (x− y)µb+ (x− y)τc).17
Then G(A, c) = (Z3×A, ◦) is a connected quandle, called the Galkin quandle correspondingto the pointed group (A, c). It is affine iff 3A = 0. It is latin iff |A| is odd. Two Galkinquandles are isomorphic iff the corresponding pointed groups are isomorphic. See [5] fordetails.
size RMlt(Q) construction properties
6 S4 6.2 or G(Z3, 0)6 S4 6.3 or 6.8 or G(Z3, 1)8 SL2(3) 6.6
7 o Z3) o Z2 G(Z7, 1) latin21 S7 6.2 simple21 PSL3(2) 6.7 simple
...33 (Z2
11 o Z3) o Z2 G(Z11, 0) latin33 (Z2
11 o Z3) o Z2 G(Z11, 1) latin
Table 2. All connected non-affine quandles of certain orders.
Table 2 lists all connected non-affine quandles of orders n ≤ 15 and n ∈ {21, 33}. Inthe column labeled “construction” we either give a reference to a numbered example whichuniquely determines the quandle, or we specify how the quandle can be constructed asGal(G,H, f) of Construction 3.1, or we specify how the quandle can be constructed asG(A, c)of Example 6.9. Note that only one quandle on 12 elements lacks detailed description.
Problem 6.10. Let p ≥ 11 be a prime. Is it true that the only non-affine connected quandlesof order 3p are the Galkin quandles G(Zp, 0) and G(Zp, 1)?
18
7. Connected quandles of size p, p2 and 2p
Let p be a prime. In this section we study connected quandles of order p, p2 and 2p.First we give two new, conceptually simple proofs for the result of Etingof, Soloviev andGuralnick [11] that connected quandles of order p are affine: the first proof uses Joyce’scharacterization of RMlt(Q) for simple quandles, the second proof requires Galois’ result onsolvable primitive groups. However, all three proofs still rely on a deep result on conjugacyclasses of prime power order by Kazarin [25]. We then mention the result of Grana [16] thatconnected quandles of order p2 are affine. We conclude with a new, purely group-theoreticalproof (modulo Theorem 4.3) of the recent result of McCarron [33] that there are no connectedquandles of order 2p if p > 5.
Proposition 7.1. Let Q be a connected rack. For a, b ∈ Q we write a ∼ b iff Ra = Rb.Then ∼ is an equivalence relation on Q, and all equivalence classes of ∼ have the samecardinality.
Proof. It is clear that ∼ is an equivalence relation. Let [a], [c] be two equivalence classesof ∼. Since Q is connected, there is θ ∈ RMlt(Q) such that aθ = c. If a ∼ b thenRc = Raθ = R θ
a = R θb = Rbθ , thus c ∼ bθ, showing that [a]θ ⊆ [c]. Since θ is one-to-one, we
deduce |[a]| ≤ |[c]|. The mapping θ−1 ∈ RMlt(Q) gives the other inequality. �
Proposition 7.2. Let Q be a connected quandle of prime power order. Then RMlt(Q) is asolvable group.
Proof. Kazarin proved in [25] that in a group G, if x ∈ G is such that |xG| is a prime power,then the subgroup 〈xG〉 is solvable.
Let Q be a connected quandle of prime power order, let G = RMlt(Q), and let ζ = Re
for any e ∈ Q. For every α ∈ G, we have ζα = Rαe = Reα . For x ∈ Q, taking α ∈ G
such that eα = x, we obtain ζα = Rx. Hence ζG = {Rx : x ∈ Q} and thus 〈ζG〉 = G. ByProposition 7.1, |ζG| is a divisor of |Q|, hence a prime power. Kazarin’s result then impliesthat 〈ζG〉 = G is solvable. �
Recall that a quandle Q is simple if its only congruences are Q×Q and {(x, x) : x ∈ Q}.
Theorem 7.3 ([11]). Every connected quandle of prime order is affine.
Proof. Let Q be the quandle in question. By Proposition 7.2, G = RMlt(Q) is solvable.Moreover, since G acts transitively on a set of prime size, it must act primitively.
Proof 1: Consequently, the quandle Q is simple, because every congruence of Q is invariantwith respect to the action of G. An observation by Joyce [23, Proposition 3] says that if Qis simple then G′ is the smallest normal subgroup in G. Since G is solvable, we then musthave G′′ = 1, hence G′ is abelian, and so Q is affine by Theorem 5.3.
Proof 2: A theorem of Galois says that a solvable primitive group acts as a subgroup ofthe affine group over a finite field. Theorem 5.3 now concludes the proof. �
Grana proved:
Theorem 7.4 ([16]). Let p be a prime. Every connected quandle of order p2 is affine.
We now turn our attention to order 2p. For every integer n ≥ 2, Example 6.2 yields aconnected quandle of order
(n2
). With n = 4 and n = 5 we obtain connected quandles of
19
order 6 = 2 · 3 and 10 = 2 · 5, respectively. These examples are sporadic in the sense that(n2
)is equal to 2p for a prime p if and only if n ∈ {4, 5}. Indeed, McCarron proved:
Theorem 7.5 ([33]). There is no connected quandle Q of order 2p for a prime p > 5.
We conclude the paper with a new, shorter proof of Theorem 7.5.Suppose that Q is a connected quandle of order 2p. Then G = RMlt(Q) ≤ S2p, G
′ actstransitively on Q by Proposition 2.3, and 〈ζG〉 = G for some ζ ∈ Z(Ge) by Theorem 4.3,so, in particular, 〈Z(Ge)
G〉 = G. Theorem 7.5 therefore follows from the group-theoreticalTheorem 8.1 below that we prove separately.
8. A result on transitive groups of degree 2p
Theorem 8.1. Let p > 5 be a prime. There is no transitive group G ≤ S2p satisfying bothof the following conditions:
(A) G′ is transitive on {1, . . . , 2p}.(B) 〈Z(G1)
G〉 = G.
We start with two general results on the center of the stabilizer of almost simple primitivegroups of degree p and 2p. Both proofs are based on the explicit classification of almostsimple primitive groups of degree p and 2p [40] (which are essentially results from [17, 30]).In the next subsection, we prove Theorem 8.1.
We will use repeatedly the easy fact that a nontrivial normal subgroup of a transitivegroup does not have fixed points.
8.1. Almost simple primitive groups of degree p, 2p.
Theorem 8.2. Let p ≥ 5 be a prime, G ≤ Sp an almost simple primitive group, U = G1
and V ≤ U with [U : V ] ≤ 2. Then Z(V ) = 〈1〉.
An explicit classification of these groups is given in [40, Lemma 3.1]:
Lemma 8.3. Let p be a prime and G ≤ Sp be an almost simple primitive group. ThenK = Soc(G) is one of the following groups:
(i) K = Ap,(ii) K = PSLd(q) acting on 1-spaces or hyperplanes of its natural projective space, d is a
prime and p = (qd − 1)/(q − 1),(iii) K = PSL2(11) acting on cosets of A5,(iv) K = M23 or K = M11.
For case (ii) we note the following fact:
Lemma 8.4. Let d ≥ 2 and q be a prime power such that (d, q) 6= (2, 2). Let G =Aut(PSLd(q)), U be the stabilizer in G of a 1-dimensional subspace, W = U ∩ PSLd(q)and V ≤ W with [W : V ] ≤ 2. Then CU(V ) = 〈1〉.
Proof. Since the graph automorphism of PSLd(q) swaps the stabilizers of 1-dimensional sub-spaces with those of hyperspaces it cannot be induced by U . Thus U ≤ PΓLd(q) and elementsof U can be represented by pairs [field automorphism, matrix] of the form[
τ,
(a 0B A
)]20
with a ∈ F∗q, B ∈ Fd−1q and A ∈ GLd−1(q) and τ ∈ 〈σ〉. Two such elements multiply as[τ1,
(a1 0B1 A1
)]·[τ2,
(a2 0B2 A2
)]=
[τ1τ2,
(a1a2 0
Bτ21 + Aτ21 B2 Aτ21 A2
)]Elements of W will have a trivial field automorphism part and a · det(A) = 1, thus the
A-part includes all of SLd−1(q). If V 6= W we have V CW of index 2, so it has a smallerA-part. (If it had a smaller B-part, this would have to be a submodule for the naturalSLd−1(q)-module which is irreducible.) The A-part cannot be smaller if d − 1 ≥ 3, or ifd− 1 = 2 and q ≥ 4.
In the remaining cases (d − 1 = 2 and q ∈ {2, 3}; respectively d − 1 = 1) the A-part canbe smaller by index 2. However we note by inspection that there is no B-part that is fixedby all A-parts by multiplication.
We now consider a pair of elements, the second being in V and the first being in CU(V ).By the multiplication formula the elements commute only if Bτ2
1 + Aτ21 B2 = Bτ12 + Aτ12 B1.
We will select elements of V suitably to impose restrictions on CU(V ).If A1 is not the identity we can set A2 as identity, B2 a vector defined over the prime field
moved by A1, and τ2 = 1 violating the equality. Similarly, if B1 is nonzero (with trivial A1)we can choose B2 to be zero, τ2 = 1 and A2 a matrix defined over the prime field that movesB1 (we noted above such matrices always exist in V ) to violate the equality. Finally, if B1
is zero and A1 the identity but τ1 nontrivial we can chose τ2 to be trivial and B2 a vectormoved by τ1 and violate the equation. This shows that the only element of U commutingwith all of V is the identity. �
Corollary 8.5. Let PSLd(q) ≤ G ≤ Aut(PSLd(q)), U be the stabilizer in G of a 1-dimensional subspace, and W ≤ U with [U : W ] ≤ 2. Then Z(W ) = 〈1〉.Proof. As subgroups of index 2 are normal we know that there exists a subgroup V ≤ W asspecified in Lemma 8.4. But then by this lemma
Proof of Theorem 8.2. For case (i) of Lemma 8.3, we have that U ∈ {Sp−1, Ap−1} and so alsoV ∈ {Sp−1, Ap−1}, thus (as p ≥ 5) clearly Z(V ) = 〈1〉. For case (ii) we get from Corollary 8.5that Z(V ) = 〈1〉. Finally for the groups in cases (iii) and (iv) an explicit calculation in GAP(as U/V is abelian we can find all candidates for V by calculating in U/U ′) establishes theresult. �
Now we turn to the case 2p.
Theorem 8.6. Let p > 5 be a prime and G ≤ S2p a primitive group. Then Z(G1) = 〈1〉.By the O’Nan-Scott theorem [29], G must be almost simple. An explicit classification of
these groups is given in [40, Theorem 4.6].
Lemma 8.7. Let p be a prime and G ≤ S2p be a primitive group. Then K = Soc(G) is oneof the following groups:
(i) K = A2p,(ii) p = 5, K = A5 acting on 2-sets,
(iii) 2p = q + 1, q = r2a
for an odd prime r, K = PSL2(q) acting on 1-spaces,21
(iv) p = 11, K = M22.
Proof of Theorem 8.6. In case (i) of Lemma 8.7 we have that G ∈ {S2p, A2p} and thusG1 ∈ {S2p−1, A2p−1} for which the statement is clearly true. Case (ii) is irrelevant hereas p = 5. Case (iii) follows from Corollary 8.5. Case (iv) is again done with an explicitcalculation in GAP. �
8.2. Proof of Theorem 8.1. We start by discussion what block systems are afforded by G.
Lemma 8.8. If G is primitive, then condition (B) is violated.
Proof. This is a direct consequence of Theorem 8.6. �
Lemma 8.9. If G affords a block system with blocks of size p, then condition (A) is violated.
Proof. Consider a block system with two blocks of size p and ϕ : G→ S2 the action on theseblocks. Then [G : Ker(ϕ)] = 2, and thus G′ ≤ Ker(ϕ) is clearly intransitive. �
So it remains to check the case when G has p blocks of size 2. Denote the set of blocks byB, let 1 ∈ B ∈ B. Labeling points suitably, we can assume that B = {1, 2}. Let S = G1 bea point stabilizer and T = GB a (setwise) block stabilizer.
Let ϕ : G → Sp be the action on the blocks. We set H = Im(ϕ) ≤ Sp and M = Ker(ϕ)and note that M ≤ Cp
2 is either trivial or has exactly p orbits of length 2.
Lemma 8.10. If M 6= 〈1〉 then T = MS.
Proof. If M 6= 〈1〉, then M has orbits of length 2. Consider t ∈ T . If 1t 6= 1 then 1t = 2 isin the same M -orbit. Thus there exists m ∈M such that 1t = 1m, thus tm−1 ∈ S. �
As p is a prime, H is a primitive group. By the O’Nan-Scott theorem [29], we know thatH is either of affine type or almost simple.
Lemma 8.11. If H is almost simple, then condition (B) is violated.
Proof. If M 6= 〈1〉 then by Lemma 8.10 Sϕ = Tϕ = H1. But then Z(S)ϕ ≤ Z(H1) = 〈1〉 byTheorem 8.6. Thus Z(S) ≤ Ker(ϕ)CG and 〈Z(S)G〉 6= G.
If M = 〈1〉 then ϕ is faithful and G ' H. The point stabilizer S ≤ G is (isomorphic to)a subgroup of the point stabilizer of H of index 2. But then by Theorem 8.2 we have thatZ(S) = 〈1〉 and thus 〈Z(S)G〉 6= G. �
It remains to consider the affine case, i.e. H ≤ Fp o F∗p. We can label the p points onwhich H acts as 0, . . . , p − 1, then the action of the Fp-part is by addition, and that of theF∗p-part by multiplication modulo p. Without loss of generality assume that Tϕ = H1. Wemay also assume that H is not cyclic as otherwise H ′ = 〈1〉 and thus G′ ≤M and condition(A) would be violated.
For p = 7 an inspection of the list of transitive groups of degree 14 [6] shows that there isno group of degree 14 which fulfills (A) and (B). Thus it remains to consider p > 7.
Let L = S ∩M = M1.
Lemma 8.12. If |L| ≤ 2 and p > 7 then condition (A) is violated.22
Proof. If |L| ≤ 2 then |M | ≤ 4 and |G| divides 4p(p − 1). Consider the number n of p-Sylow subgroups of G. Then n ≡ 1 (mod p) and n divides 4(p− 1). Thus n = ap + 1 witha ∈ {0, 1, 2, 3} and b(ap + 1) = 4(p − 1). If a 6= 0 this implies that b ∈ {1, 2, 3, 4}. Tryingout all combinations (a, b) we see that there is no solution for a > 0, p > 7.
So n = 1. But a normal p-Sylow subgroup must have two orbits of length p, which asorbits of a normal subgroup form a block system for G. The result follows by Lemma 8.9. �
This in particular implies that we can assume that M 6= 〈1〉, thus by Lemma 8.10 we havethat Sϕ = H1 ≤ F∗p. Thus there exists b ∈ S such that H1 = 〈bϕ〉.
Lemma 8.13. S = 〈b〉 · L.
Proof. Clearly S ≥ 〈b〉 · L. Consider s ∈ S. Then sϕ ∈ H1, thus sϕ = (bϕ)x for a suitable xand thus sb−x ∈ Ker(ϕ) ∩ S = L. �
We shall need a technical lemma about finite fields. For β ∈ F∗p, a subset I ⊂ Fp is calledβ-closed if Iβ = I, that is x ∈ I ⇔ xβ ∈ I.
Lemma 8.14. Let α, β ∈ F∗p, β 6= 1 and assume that ∅ 6= I ⊂ F∗p is β-closed. ThenI − α = {i− α | i ∈ I} is not β-closed.
Proof. Assume that I − α is β-closed and consider an arbitrary x ∈ I. Then (as β has afinite multiplicative order) xβ−1 ∈ I and thus xβ−1 − α ∈ I − α. But by the assumption(xβ−1 − α)β ∈ I − α and thus (xβ−1 − α)β + α = x + α(1 − β) ∈ I. Thus I would beclosed under addition of α(1− β) 6= 0. But the additive order of a nonzero element in Fp isp, implying that I = Fp, contradicting that 0 6∈ I. �
Lemma 8.15. If condition (A) holds, then Z(S) ≤ L ≤M .
Proof. Assume the condition holds. We show the stronger statement that CS(L) ≤ L. Forthis assume to the contrary that bx · l ∈ CS(L) with l ∈ L and x a suitable exponent suchthat bx 6∈ L. As L ≤ M is abelian this implies that bx ∈ CS(L). Let β ∈ F∗p ≤ H be suchthat (bx)ϕ = β. As bx 6∈ L we know that β 6= 1.
When we consider the conjugation action of G on M ≤ Cp2 , note that an element of M is
determined uniquely by its support (that is the blocks in B whose points are moved by theelement), which we consider as a subset of Fp, which is the domain on which H acts. Anelement g ∈ G acts by conjugation on M with the effect of moving the support of elements inthe same way as gϕ moves the points Fp. For bx to centralize an element a ∈ L, the supportI of a thus must be β-closed for β = (bx)ϕ.
By Lemma 8.12 we can assume that |L| > 2. Thus there exists an element a ∈ L whosesupport I is a proper nonempty subset of F∗p. Thus there exists α ∈ F∗p, α 6∈ I.
That means that if we conjugate a with −α ∈ Fp, the resulting element a has supportI − α. By assumption 0 6∈ I − α, so a ∈ L. But by Lemma 8.14 we know that I − α is notβ-closed, that is a ∈ L is not centralized by bx. �
Corollary 8.16. If H is of affine type, then at least one of conditions (A), (B) is violated.
Proof. If (A) holds, then 〈Z(G1)G〉 ≤M 6= G. �
This concludes the proof of Theorem 8.1.23
References
[1] N. Andruskiewitsch and M. Grana, From racks to pointed Hopf algebras, Adv. Math. 178 (2003), no.2, 177–243.
[2] V.D. Belousov, Osnovy teorii kvazigrupp i lup, (Russian) [Foundations of the theory of quasigroups andloops] Izdat. “Nauka”, Moscow, 1967.
[3] E. Brieskorn, Automorphic sets and singularities, Contemp. Math. 78 (1988), 45–115.[4] R.H. Bruck, A survey of binary systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer
Verlag, Berlin-Gottingen-Heidelberg, 1958.[5] W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito and Timothy Yeatman, Con-
nected quandles associated with pointed abelian groups, Pacific J. Math. 264 (2013), no. 1, 31–60.[6] J. H. Conway, A. Hulpke, and J. McKay, On transitive permutation groups, LMS J. Comput. Math. 1
(1998), 1–8.[7] H.S.M. Coxeter, Regular Polytopes, Courier Dover Publications, New York, 1973[8] L.E. Dickson, Linear groups: With an exposition of the Galois field theory, Dover, New York, 1958.[9] V.G. Drinfeld, On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad,
1990), Lecture Notes in Math. 1510, Springer-Verlag, Berlin, 1992, 1–8.[10] G. Ehrman, A. Gurpinar, M. Thibault and D.N. Yetter, Toward a classification of finite quandles, J.
Knot Theory Ramifications 17 (2008), no. 4, 511–520.[11] P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang-
Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001), no. 2, 709–719.[12] V.M. Galkin, Left distributive finite order quasigroups (Russian), Mat. Issled. No. 51 (1979), 43–54.[13] V.M. Galkin, Quasigroups (Russian), translated in J. Soviet Math. 49 (1990), no. 3, 941–967. Itogi
Nauki i Tekhniki, Algebra. Topology. Geometry, Vol. 26 (Russian), 3–44, 162, Akad. Nauk SSSR,Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988.
[14] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.3; 2013. (http://www.gap-system.org)
[15] G. Glauberman, On loops of odd order, J. Alg. 1 (1964), 374–396.[16] M. Grana, Indecomposable racks of order p2, Beitrage Algebra Geom. 45 (2004), no. 2, 665–676.[17] R. M. Guralnick, Subgroups of prime power index in a simple group, J. Algebra 81 (1983), no. 2,
304–311.[18] H. Holmes, D. Stanovsky, Affine quandles revisited, in progress.[19] X. Hou, Finite modules over Z[t, t−1], J. Knot Theory Ramifications 21 (2012), no. 8, 1250079, 28 pp.[20] A. Hulpke, Constructing transitive permutation groups, J. Symbolic Comput. 39 (2005), no. 1 (2001),
1–30.[21] P. Jedlicka, A. Pilitowska, D. Stanovsky and A. Zamojska-Dzienio, The structure of medial quandles,
submitted.[22] D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Alg. 23 (1982), 37–66.[23] D. Joyce, Simple quandles, J. Algebra 79 (1982), 307–318.[24] M. Kano, H. Nagao and N. Nobusawa, On finite homogeneous symmetric sets, Osaka J. Math. 13 (1976),
Math. Notes 48 (1990), 749–751.[26] T. Kepka and P. Nemec, Commutative Moufang loops and distributive groupoids of small orders,
Czechoslovak Math. J. 31(106) (1981), no. 4, 633–669.[27] M. Kikkawa, On some quasigroups of algebraic models of symmetric spaces, Mem. Fac. Lit. Sci., Shimane
Univ. (Natur. Sci.) 6 (1973), 9–13.[28] D. Larue, Left-distributive idempotent algebras, Commun. Algebra 27/5 (1999), 2003–2009.[29] M. W. Liebeck, C. E. Praeger, and J. Saxl, On the O’Nan-Scott theorem for finite primitive permutation
groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389–396.[30] M. W. Liebeck and J. Saxl, Primitive permutation groups containing an element of large prime order,
J. London Math. Soc. (2) 31 (1985), no. 2, 237–249.[31] O. Loos, Symmetric spaces, J. Benjamin New York, 1969.
24
[32] S. V. Matveev, Distributive groupoids in knot theory, Math. USSR - Sbornik 47/1 (1984), 73–83.[33] J. McCarron, Connected quandles with order equal to twice an odd prime,
http://arxiv.org/abs/1210.2150[34] G. Murillo, S. Nelson and A. Thompson, Matrices and finite Alexander quandles, J. Knot Theory
Ramifications 16 (2007), no. 6, 769–778.[35] H. Nagao, A remark on simple symmetric sets, Osaka J. Math. 16 (1979), 349–352.[36] S. Nelson, C.-Y. Wong, On the orbit decomposition of finite quandles, J. Knot Theory Ramifications 15
(2006), no. 6, 761–772.[37] N. Nobusawa, Some structure theorems on pseudo-symmetric sets, Osaka J. Math. 20 (1983), 727–734.[38] N. Nobusawa, Jordan-Holder theorem for pseudo-symmetric sets, Osaka J. Math. 23 (1986), 853–858.[39] R. S. Pierce, Symmetric groupoids, Osaka J. Math. 15/1 (1978), 51–76.[40] J. Shareshian, On the Mobius number of the subgroup lattice of the symmetric group, J. Combin. Theory
![Engineering the Internet of Things€¦ · nected devices globally, representing a market opportunity approaching $11 trillion [1]. While innovations in factory automation, smart](https://static.documents.pub/doc/80x56/5f47c14987a16c547038b851/engineering-the-internet-of-things-nected-devices-globally-representing-a-market.jpg)
