Connected Quandles and Transitive Groups 1

CONNECTED QUANDLES AND TRANSITIVE GROUPS

ALEXANDER HULPKE, DAVID STANOVSKY,´ AND PETR VOJTECHOVSKˇ Y´

Abstract. Building on ideas of Galkin, we establish a canonical representation of connected 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

Rx : Q → Q, y 7→ yx are permutations of Q. In a right quasigroup, the permutation group

RMlt(Q) = hRx : 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 GACRˇ grant 13-01832S to David Stanovský,and the Simons Foundation Collaboration Grant 210176 to Petr Vojtˇechovský. 1 into orbits of transitivity. The orbits are not necessarily connected, but they share certain properties with connected racks. In this paper we are mostly interested in connected quandles, but some of our observations apply to general quandles and racks as well. In [42, Section 4], one can find hints how to build connected racks over connected quandles.

1.3. Motivation and history of connected quandles. One of the main motivations behind the theory of quandles is finding computable knot and link invariants—the three defining 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 the same orbit of transitivity. Disconnected quandles are of importance for links. From a broader perspective, quandles are a special type of set-theoretical solutions to the quantum Yang-Baxter equation as formulated by Drinfeld [9]. There are indications, see [11], that understanding racks and quandles is an important step towards understanding general set-theoretical solutions of the quantum Yang-Baxter equation. The investigation of racks started with several special cases. A rack Q is called involutory 2 if Rx = 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 latin quandles: Medial latin quandles are affine, by the early Toyoda-Bruck theorem [4]. Latin quandles that are also left distributive are affine over commutative Moufang loops [2], and many strong properties follow from this connection; see [26, 41]. General latin quandles were studied in a series of papers by Belousov, Galkin and their collaborators; see Belousov’s book [2] and Galkin’s survey paper [13] for more information. Involutory latin quandles are essentially the same objects as Bruck loops of odd order [15, 27]. The theory of quandles is younger, particularly the theory of connected quandles. Here is a 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 ideas of [12] were rediscovered. Vendramin [45] used this representation to enumerate connected quandles up to order n ≤ 35. Different methods lead to the classification of quandles on p, 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]. Early structural results on connected involutory quandles appeared in [24, 35, 39], and more can be 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 connected quandles as certain configurations in transitive groups. Some variants of this representation were discovered independently in [10, 12, 22, 39], but none of these works contains a complete characterization of our Theorem 4.3, nor a discussion of the isomorphism problem as in our Theorem 4.6. Using the representation theory, we can prove several known results in a simpler and faster way, and also obtain new results. The paper is organized as follows. In Section 2 we develop basic properties of quandles and racks in relation to the right multiplication group and its derived subgroup. Sections 3 and 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 right multiplication group (Proposition 4.8). Section 5 contains a characterization of connected affine quandles (Theorem 5.3). We show that connected quandles are affine if and only if they are medial if and only if their right multiplication group is metabelian. In Section 6 we present an algorithm for enumerating small connected quandles that is similar 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 and geometric methods, we construct several families of connected quandles. Thanks to Theorem 4.3, 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. Using Theorem 4.3, we first show that if Q is a connected quandle of prime power order then RMlt(Q) is solvable (Proposition 7.2). We then give two conceptually simple proofs of the fact 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 prime power 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 of quandles to show that there are no connected quandles of order 2p. We give a new and shorter proof of this fact. First, in Section 7 we use Theorem 4.3 to show that McCarron’s result 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 we let [x, y] = x−1xy. Since (x−1)y = (xy)−1, we denote both of these elements by x−y. α For α ∈ Aut(G) we let CG(α) = {z ∈ G : z = z} be the centralizer of α, and we write CG(x) for CG(φx). g If G acts on X and x ∈ X, we let Gx = {g ∈ G : x = x} be the stabilizer of x, and xG = {xg : g ∈ G} the orbit of x. 3 α Note that for any binary algebra (Q, ·), every b ∈ Q and α ∈ Aut(Q) the mapping Rb is equal to Rbα , because for every a ∈ Q we have

R α α−1R α α−1 α α R α a b = a b = (a · b) = a · b = a b .

−α α −1 −1 Consequently, if Rb is a permutation, then Rb = (Rb ) = Rbα . We will use this observation 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 subgroup of the right multiplication group plays an important role. Most of the material can be found in Joyce’s papers [22, 23] or even earlier [24, 35]. For a rack Q, deﬁne the group of displacements as

−1 Dis(Q) = hRaRb : a, b ∈ Qi. Note that Dis(Q) ≤ RMlt(Q) ≤ Aut(Q).

For every a, b ∈ Q, we have R−1R ∈ Dis(Q), too, as R−1R = R R−Rb = R R−1 = R R−1. a b a b b a b aRb b ab Proposition 2.1. Let Q be a rack. Then: 0 (i) RMlt(Q) E Dis(Q) E RMlt(Q) and the group RMlt(Q)/Dis(Q) is cyclic. k1 kn Pn (ii) Dis(Q) = {Ra1 ...Ran : n ≥ 0, ai ∈ Q and 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.

−1 α Proof. (i) Let G = RMlt(Q) and D = Dis(Q). For a, b ∈ Q and α ∈ G we have (RaRb ) = α −α −1 Ra Rb = Raα Rbα ∈ D, proving D E G. Fix e ∈ Q and note that DRa = DRe for every k1 kn k1+···+kn a ∈ Q. Each element α ∈ G is of the form α = Ra1 ...Ran . Then Dα = DRe , proving 0 G/D = hDRei. 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 note k1 kn P that every α ∈ S can be written as Ra1 ...Ran , where not only i ki = 0 but also ki = ±1. Assuming such a decomposition, we prove by induction on n that α ∈ D. −1 If n = 0 then α = 1, the case n = 1 does not occur, and if n = 2, we have either α = RaRb −1 or α = R Rb, both in D. Suppose that n > 2. a P P If k1 = kn then there is 1 < m < n such that i

Proof. (i) By definition, Dis(Q) = 1 iff Ra = Rb for every a, b ∈ Q. If Q is a quandle, we then get ab = aRb = aRa = a. (ii) Note that the following identities are equivalent: Q is medial, RyRuv = RuRyv, −1 −1 RyRv RuRv = RuRv RyRv, −1 −1 (2.1) RyRv Ru = RuRv Ry. −1 −1 −1 −1 Suppose that Dis(Q) is commutative. Then (RyRv )(RuRy ) = (RuRy )(RyRv ) = −1 RuRv , which yields (2.1) upon applying Ry to both sides. Hence Q is medial. −1 −1 −1 −1 Conversely, if Q is medial, then (2.1) holds, and its inverse yields Ry RvRu = Ru RvRy , −1 −1 −1 −1 −1 −1 so RxRy RvRu = RxRu RvRy = RvRu RxRy , 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 (or Alexander) 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 (x ∗ y) ∗ (u ∗ v) = (xf + y1−f ) ∗ (uf + v1−f ) = xf 2 + y(1−f)f + uf(1−f) + v(1−f)2 is invariant under the interchange of y and u. 5 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. (Without loss 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 later that 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 quandles based on transitive permutation groups: the minimal representation of Theorem 3.5 and the canonical 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 aﬃne quandles 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)f y. First we note that the operation ∗ is well deﬁned. Indeed, if Hx = Hu and Hy = Hv then u = hx, v = ky for some h, k ∈ H, and H(uv−1)f v = H(hxy−1k−1)f ky = Hhf (xy−1)f (k−1)f ky = Hh(xy−1)f k−1ky = H(xy−1)f y, using H ≤ CG(f). In fact, Gal(G, H, f) is always a quandle. Idempotence is immediate from Hx ∗ Hx = H(xx−1)f x = Hx. For right distributivity we calculate (Hx ∗ Hz) ∗ (Hy ∗ Hz) = H(xz−1)f z ∗ H(yz−1)f z = H[(xz−1)f z((yz−1)f z)−1]f (yz−1)f z = H(xy−1)f 2 (yz−1)f z = H((xy−1)f yz−1)f z = H(xy−1)f y ∗ Hz = (Hx ∗ Hy) ∗ Hz. It remains to check that right translations in Gal(G, H, f) are permutations of G/H. Note that for x, y, z ∈ G we have Hx ∗ Hy = Hz ⇔ H(xy−1)f y = Hz ⇔ Hxf (yf )−1 = Hzy−1 ⇔ Hxf = Hzy−1yf ⇔ Hx = H(zy−1)f −1 y,

−1 where in the last step we applied f 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 abelian group 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 be the 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 representable. 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

G α (3.2) πe : Gal(G, Ge, f) → e ,Geα 7→ e .

α β Since Geα = Geβ holds if and only if e = e , 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 group RMlt(Q) or the displacement group Dis(Q). Let f be the restriction of the conjugation by Q Re in RMlt(Q) onto G. Then Gal(G, Ge, f) is well deﬁned and isomorphic to the orbit e .

Proof. Since f is a restriction of the conjugation by Re ∈ RMlt(Q) onto a normal subgroup G 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

−1 f −1 −1 Geα ∗ Geβ = Ge(αβ ) β = GeRe αβ Reβ. G Q Consider the bijective mapping πe from (3.2). By Proposition 2.1(iii), e = e . To see that πe is a homomorphism, we calculate

−1 −1 −1 πe Re αβ Reβ αβ β α β πe πe (Geα ∗ Geβ) = e = (e · e) = e · e = (Geα) · (Geβ) ,

R where we have used e e = e and β ∈ Aut(Q). Given a connected quandle Q and an element e ∈ Q, we will call the Galkin representation

Q ' 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)0 = Dis(Q). We will discuss canonical representations in the next section. The following result explains why we have used the adjective “minimal.” Theorem 3.5 (Minimal representation of connected quandles). Let Q be a connected quandle. Then: 0 (i) Q ' Gal(G, H, f) whenever G = RMlt(Q) , e ∈ Q, H = Ge and f = φRe is the conjugation by Re on G. (ii) If Q ' Gal(G, H, f) for some G, H ≤ G and f ∈ Aut(G), then RMlt(Q)0 embeds into a quotient of G. 7 Proof. Part (i) is just Proposition 3.4 with Q = eQ. Let us therefore assume the hypothesis of part (ii), where for simplicity we take Q = Gal(G, H, f). Deﬁne ϕ : G → Aut(Q) by ϕa a 7→ ϕa, where (Hx) = Hxa. The mappings ϕa are automorphisms of Q, since (Hx)ϕa ∗ (Hy)ϕa = Hxa ∗ Hya = H(xaa−1y−1)f ya = H(xy−1)f ya = (Hx ∗ Hy)ϕa . The mapping ϕ is obviously a homomorphism. We show that RMlt(Q)0 is a subgroup of Im(ϕ), and hence that RMlt(Q)0 embeds into G/Ker(ϕ). 0 −1 By Proposition 2.3, RMlt(Q) = Dis(Q). It therefore suﬃces to check that RHxRHy ∈ Im(ϕ) for every x, y ∈ G. Recall that the unique solution to Hx ∗ Hy = Hz is Hx = −1 R−1 −1 H(zy−1)f y, and thus (Hz) Hy = H(zy−1)f y. Hence for every x, y, u ∈ G we have

R R−1 −1 f R−1 −1 f −1 f −1 −1 −1 f −1 (Hu) Hx Hy = (H(ux ) x) Hy = H((ux ) xy ) y = Hux (xy ) y, −1 proving RHxRHy = ϕx−1(xy−1)f−1 y. Corollary 3.6. Let Q be a ﬁnite connected quandle, and let G be of smallest order among all groups such that Q ' Gal(G, H, f). Then G ' RMlt(Q)0.

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 configurations in 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 hζGi = G, that is, if the smallest normal subgroup 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 calculate αRe α α α α Reα x = x · e = x · e = (xe) = x , so Re ∈ Z(Ge). Since Q is connected, G acts xb xb transitively on Q, and for every x ∈ Q there is x ∈ G such that e = x. Then Re = Rexb = Rx, G b proving hRe i = G. For a quandle folder (G, ζ), define Q(G, ζ) = (Q, ◦), x ◦ y = xζyb, y where yb is any element of G satisfying eb = y. We shall see that the operation does not depend on the choice of the permutations yb, 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 yb. (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)) = hζyb : y ∈ Qi = hζGi. (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 condition α β certainly holds when e = e because ζ ∈ Z(Ge). This proves (i), and part (ii) follows. Consider again the bijection πe of (3.2). Since G is transitive, πe is onto Q. To check that πe β ecβ is a homomorphism, note that ζ = ζ by (i). Therefore, with Gal(G, Ge, φζ ) = (G/Ge, ∗), −1 ζ −1 β β we have Geα ∗ Geβ = Ge(αβ ) β = Geζ αζ = Geαζ , and thus

β ecβ πe β πe αζ α ζ α β πe πe (Geα ∗ Geβ) = (Geαζ ) = e = (e ) = e ◦ e = (Geα) ◦ (Geβ) . This proves (iii), and part (iv) follows. For (v), note that the right translation by y in (Q, ◦) is the mapping ζyb and, once again, β ecβ ζ = ζ 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 = xζyb are mutually inverse bijections between the set of connected quandles and the set of quandle envelopes on the set Q. Proof. In view of Lemmas 4.1 and 4.2, it remains to show that the two mappings are mutually inverse. Let (G, ζ) be a quandle envelope, and let (Q, ◦) = Q(G, ζ) be the corresponding connected quandle. Then RMlt(Q, ◦) = hζGi = G by Lemma 4.2. Moreover, xRe = x ◦ e = ζeb ζ x = x thanks to eb ∈ Ge and ζ ∈ Z(Ge), hence ζ is the right translation by e in (Q, ◦). It follows that E(Q(G, ζ)) = (G, ζ). Conversely, let Q be a connected quandle, and E(Q) = (RMlt(Q),Re) the corresponding y quandle envelope. Then, in Q(E(Q)), we calculate x ◦ y = xReb = 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 acts transitively on QK , and the stabilizer of a fixed element e ∈ Q is the peripheral subgroup HK . Since HK ' Z × Z, the meridian mK is central in the stabilizer, and it follows from GK Wirtinger’s presentation of GK that GK = hmK i. We proved that (GK , mK ) is a quandle envelope, 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 representations. We will take advantage of this result in Algorithm 6.1, where we enumerate all connected quandles of given size up to isomorphism. We start with a useful characterization of 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; ψ ψ (iii) C be the set of all group isomorphisms ψ : G → K such that ζ = ξ and Ge = 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 maps B into C, and construct a mapping g : C → A ⊆ B such that fg is the identity mapping on B and gf is the identity mapping on C. ζyb y Let Q(G, ζ) = (Q, ◦), where x ◦ y = x for some yb ∈ G satisfying eb = y, and Q(K, ξ) = (Q, ∗), where x ∗ y = xξy for some y ∈ K such that ey = y. Note that the following identities are equivalent for a permutation ϕ of Q: ϕ ϕ (x ◦ y)ϕ = (xϕ) ∗ (yϕ), (xζyb)ϕ = (xϕ)ξy , xϕ−1ζybϕ = xξy . Hence ϕ is an isomorphism (Q, ◦) → (Q, ∗) if and only if

ϕ (ζyb)ϕ = ξy . 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 = (ζeb)ϕ = ξeϕ = ξe = ξ. To prove Gϕ ⊆ K, note that G = hζGi, pick α ∈ G, and calculate (ζα)ϕ = (ζecα )ϕ = ξeαϕ ∈ K. For the other inclusion K ⊆ Gϕ, note that K = hξK i, pick β ∈ K, ﬁnd α ∈ G such that eβ = eαϕ by transitivity of G, and calculate ξβ = ξeβ = ξeαϕ = (ζecα )ϕ ∈ Gϕ. f ϕ (f : B → C): For ϕ ∈ B let ψ = ϕ = φϕ be the conjugation by ϕ. Since G = K, we see ψ ϕ ψ that ψ is an isomorphism G → K. Clearly ζ = ζ = ξ. To verify Ge = Ke, let α ∈ Ge and αψ αϕ ϕ−1αϕ ψ calculate e = e = e = e, so α ∈ Ke. (g : C → A): For ψ ∈ C, deﬁne ϕ = ψg by

ψ xϕ = exb for every x ∈ Q. We show that ϕ is an isomorphism (Q, ◦) → (Q, ∗) that ﬁxes e. The second ψ ϕ eb ψ condition follows immediately from e = e = e, because eb ∈ Ge and Ge = Ke. Let us observe two facts. First, if α, β ∈ G, then the following conditions are equivalent: αψ βψ βψ(αψ)−1 −1 ψ −1 α β e = e , e = e, (βα ) ∈ Ke, βα ∈ Ge, e = e .

α This implies that ϕ is a bijection. Second, for any x ∈ Q and α ∈ G we have exc = xα = exαb . Combining the two observations, we see that

αψ ψ (4.1) exc = e(xαb ) . For x, y ∈ Q, we then have ψ ψ y y ψ ψ y ψ (x ◦ y)ϕ = e[x◦y = exζdb = e(xζb b) = exb (ζ b)

ψ ψ ϕ = (xϕ)(ζyb)ψ = (xϕ)(ζψ)(yb ) = (xϕ)ξ(yb ) = (xϕ)ξy = xϕ ∗ yϕ,

ψ where in the penultimate step we used eyb = yϕ. (fg = id): For ϕ ∈ B and x ∈ Q we have

fg f g (ϕf ) ϕ −1 xϕ = x(ϕ ) = exb = exb = eϕ xϕb = exϕb = 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

g ψ g −1 g g αψ ψ ψ ψ g ψ ψ xα = x(ψ ) αψ = (uα)ψ = euc = e(uαb ) = eub α = (uψ )α = xα . 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. ψ ψ (iii) There is an isomorphism ψ : G → K such that ζ = ξ and Ge = 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 = ρα is also an isomorphism Q(G, ζ) → Q(K, ξ) and it satisfies eϕ = e. The rest follows from Lemma 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 choice of 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 a connected quandle in terms of its right multiplication group. For a group G, a subgroup H ≤ G and an element x ∈ G we let ψ ψ Aut(G)x,H = {ψ ∈ Aut(G): x = x, H = H} ≤ Aut(G). Proposition 4.8. Let Q be a connected quandle, e ∈ Q, and let G = RMlt(Q). Then −1 Aut(Q) is isomorphic to (G o Aut(G)Re,Ge ) /{(α, φα ): α ∈ Ge}.

Proof. By Theorem 4.3, we have Q = Q(G, Re). According to Lemma 4.5, ϕ 7→ φϕ is a bijection between Aut(Q)e and Aut(G)Re,Ge , which is easily seen to be a homomorphism. f Define f : G o Aut(Q)e → Aut(Q) by (α, ϕ) = 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, where α ∈ G and ϕ ∈ Aut(Q)e. Thus f is surjective. The kernel of f consists of all tuples (α, ϕ) −1 with αϕ = 1, hence ϕ = α ∈ 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 affine mappings 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) ∈ A o 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 ,

−1 −1 −1 xRy = xf + y1−f , 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 use on 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, +). It suffices 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) −1 then also xRy , xRy ∈ Im(1 − f). If x = u1−f ∈ Im(1 − f) then xRy = u(1−f)f + y1−f = −1 −1 −1 −1 (uf + y)1−f ∈ Im(1 − f), and xRy = u(1−f)f + y1−f = (y − u)1−f ∈ 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). It follows 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 if and 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 affine if and only if it is medial. (Recall that there are medial quandles that are not affine, as illustrated by Example 2.2.) In the finite case, the result follows from [5, Theorem 5.10] mentioned above, and from the Toyoda-Bruck theorem [4] that states that medial quasigroups are affine. Our method is substantially different, includes the infinite case, and provides a new proof of a special case of the Toyoda-Bruck theorem for idempotent medial quasigroups (it does not extend to non-idempotent quasigroups in a straightforward fashion). The crucial point in Theorem 5.3 is characterization (iv), which is interesting on its own and 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)0 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 is medial. By Propositions 2.3 and 2.4, every connected medial quandle Q has RMlt(Q)0 = Dis(Q) abelian. 12 (iii) ⇒ (iv): Fix e ∈ Q. Since RMlt(Q)0 is abelian and transitive (by Propositions 2.1 and 0 2.3), it is sharply transitive. Thus for every y ∈ Q there is a unique yb ∈ RMlt(Q) such that eyb = y. Define A = (Q, +) by x + y = xyb. 0 We claim that ϕ : A → RMlt(Q) , x 7→ xb is an isomorphism and hence that A is an abelian xcyb y xy y group. Indeed, ϕ is clearly a bijection, we have e = xb = ebb, thus xbb = xbyb by sharp ϕ y ϕ ϕ ϕ transitivity, and then (x + y) = (xb) = xbyb = xy = x y . bb 0 0 Since the right translation by y in A is yb ∈ 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) and Re 1−Re x · y = x + y , because then Ry ∈ Aff(A) for every y ∈ Q. We have Re ∈ Aut(A) R iff (x + y)Re = xyRb e is equal to xRe + yRe = xReyde for every x, y ∈ Q, which is equivalent Re R to yb = yce for every y ∈ Q. Taking advantage of sharp transitivity, the last equality is Re R verified by eyb = y · e = eyde . We have Q = Q(E(Q)) by Theorem 4.3, and hence

yb −1 x · y = xRe = xyb Reyb = (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 yb right translation by y in A. By Theorem 4.3, we have Ry = R0 for some yb ∈ RMlt(Q) such that 0yb = y. Since RMlt(Q) ≤ Aﬀ(A), there are c ∈ Q and g ∈ Aut(A) such that xyb = c+xg y y g for every x ∈ Q. But c = 0b = y, so xb = y + x and y = gρy. Since Mlt(A) ≤ RMlt(Q), −1 b we have g = yρb y ∈ RMlt(Q). Hence g ∈ RMlt(Q)0, and since R0 ∈ Z(RMlt(Q)0), we R0 obtain gR0 = R0g. Since 0 = 0 by idempotence, we have not only R0 ∈ Aﬀ(A) but in fact R0 ∈ Aut(A). Using all these facts, we calculate

R yb y−1R y ρ−1g−1R gρ ρ−1R ρ R R 1−R x · y = x 0 = xb 0 b = x y 0 y = x y 0 y = (x − y) 0 + y = x 0 + y 0 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 enumeration of 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 mapping ϕ : Im(1 − f) → Dis(Q) which maps c to the translation by c is an isomorphism. Note that T is a group. (Dis(Q) ⊆ T ): We calculate

−1 −1 −1 −1 −1 zRxRy = (zf + x1−f )f + y1−f = z + x(1−f)f + y1−f ,

−1 −1 −1 so zRxRy = z + c with the constant c = x(1−f)f + y1−f ∈ 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 R R−1 f 1−f f −1 z x 0 = (z + x ) = z + c. Corollary 5.5. Let (A, ∗) = Aff(A, f) be a connected affine quandle. Then the isomorphism type 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) for any f ∈ Aut(A), g ∈ Aut(B) with 1 − f and 1 − g onto. In general, an affine quandle that is 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 its Cayley table, is affine, and when it is, how to find its affine representation. They provided a simple but inefficient algorithm based on a couple of necessary conditions and a brute force search. We note that for a connected quandle Q the problem is rather easy, thanks to Theorem 5.3, as it suffices to test whether G = RMlt(Q)0 is abelian, and in the positive case return A = G and f = φRe for any e ∈ Q. An efficient test of affinity for general quandles will be presented in [18]. Proposition 5.7. Let Aff(A, f), Aff(A, g) be connected affine quandles. Then Aff(A, f) is isomorphic to Aff(A, g) if and only if f and g are conjugate in Aut(A). Proof. Suppose that g = f ϕ for some ϕ ∈ Aut(A). Then (xf + y1−f )ϕ = xϕf ϕ + yϕ − yϕf ϕ = (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 = f ϕ. Since 1 − f is onto A by Proposition 5.1, given x, y ∈ A there are u, v ∈ A such that x = uf and y = v1−f . Then (x + y)ϕ = (uf + v1−f )ϕ = uϕg + vϕ(1−g) = ufϕ + v(1−f)ϕ = xϕ + yϕ shows that ϕ ∈ Aut(A). Corollary 5.5 and Proposition 5.7 can be used to enumerate finite connected affine quandles up 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 it suffices 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) ' Zp such that 1 − f 6= 0, that is, f 6= 1. Since Aut(A) is abelian, conjugacy plays no role, and we obtain p − 2 connected affine quandles with p elements. In [19], Hou proved a stronger result than Proposition 5.7, solving the isomorphism problem for all finite affine quandles (not necessarily connected). Using the method described above, he found explicit formulas for the number of affine quandles up to isomorphism with pk elements, k = 1, 2, 3, 4, both in the general and the connected cases. For example, on p2 2 2 elements, there are precisely 2p − 3p − 1 connected affine quandles, p − 2p with A = Zp2 2 and p − p − 1 with A = Zp × Zp. (According to Theorems 7.3 and 7.4, all quandles with p and p2 elements are affine.) We also note that a variant of Proposition 5.7 in the setting of distributive quasigroups was proved by Kepka and Nˇemec[26]. They used it to show that non-medial distributive quasigroups exist only on 3k elements, k ≥ 4, and enumerated them for k = 4, 5. (In the quandle terminology, we speak of latin distributive quandles. Recall that all finite connected distributive 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 all quandle envelopes (G, ζ), where G is a transitive group on Q, and where ζ ∈ Z(Ge) satisfies hζGi = 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) (and ζ = Re). Propositions 2.1 and 2.3 then imply that it suffices to consider transitive groups G for which G0 is also transitive and G/G0 is cyclic. This disqualifies many transitive groups. In principle, Theorem 4.6 then solves the isomorphism problem: Given two quandle envelopes (G, ζ) and (K, ξ), the connected quandles Q(G, ζ) and Q(K, ξ) are isomorphic if and only if G is isomorphic to K, i.e., G = K if we start with a list of transitive groups up to isomorphism, 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 a direct isomorphism check on all quandles constructed from all quandle envelopes (G, ζ) with a 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 G0 is transitive and G/G0 is cyclic then qG ← ∅ G for each ζ in Z(G1) such that hζ i = 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 output of the search are available on the website of the third author. The isomorphism check is based on the methods of the LOOPS package for GAP. The current version of GAP contains a library of transitive groups up to degree 30, and an extension up to degree 35 can be obtained 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 5 minutes, 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 3.4 with G = RMlt(Q). Consequently, he had to deal with many more transitive groups, had to filter out quandles that are not connected, and also had to filter many quandles up to isomorphism, 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) of latin quandles of size n, and the number a(n) of connected affine quandles of size n, up to isomorphism. Latin quandles are recognized by a direct test whether all left translations 15 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 q(n) 1 0 1 1 3 2 5 3 8 1 9 10 11 0 7 9 15 12 `(n) 1 0 1 1 3 0 5 2 8 0 9 1 11 0 5 9 15 0 a(n) 1 0 1 1 3 0 5 2 8 0 9 1 11 0 3 9 15 0

n 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 q(n) 17 10 9 0 21 42 34 0 65 13 27 24 29 17 11 0 15 `(n) 17 3 7 0 21 2 34 0 62 7 27 0 29 8 11 0 15 a(n) 17 3 5 0 21 2 34 0 30 5 27 0 29 8 9 0 15 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 G0 is abelian, using Proposition 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 values of a(n) in Table 1 except for a(32). We conclude this section by providing examples of sequences of connected quandles. The ﬁrst source of examples is combinatorial, resulting from multitransitivity of the symmetric and alternating groups.

Example 6.2. For n ≥ 2 let G = Sn act on 2-element subsets of {1, . . . , n}, let e = {1, 2} G and ζ = (1, 2). Then ζ ∈ Z(Ge) and hζ i = G, since all transpositions are conjugate to ζ in n Sn. Thus Q(G, ζ) is a connected quandle of order 2 .

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 and G Ge = Z(Ge) = hζi, so certainly ζ ∈ Z(Ge). Furthermore, hζ i generates Sn if n is even (and An 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 have G Ge = Z(Ge) = hζi, so ζ ∈ Z(Ge), and hζ i = G. Thus Q(G, ζ) is a connected quandle of order 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) = hζi, so ζ ∈ Z(Ge). As An is generated by 3-cycles, we also have hζGi = G. Thus Q(G, ζ) is a connected quandle of order n!/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 of 2 all non-zero vectors in the plane (Fq) . Let e = (1, 0). A quick calculation shows that 16 1 0 Ge = {Ma : a ∈ Fq}, where Ma = ( a 1 ). Let ζ = M1. Since MaMb = Ma+b, we have G Ge ' (Fq, +), so ζ ∈ Z(Ge) = Ge. We claim that hζ i = 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 ∗ G q is even then Fq has odd order q − 1 and thus every element of Fq is a square, so Ge ≤ hζ i. ∗ When q is odd then Fq contains (q − 1)/2 squares, so there are at least (q − 1)/2 + 1 > q/2 G G elements in hζ i ∩ Ge conjugate to ζ, and Lagrange’s Theorem then implies that Ge ≤ hζ i again. G G G Since Ge ≤ hζ i, we establish hζ i = G by proving that hζ i acts transitively on Q. −1 0 1 −1 Given (x, y) ∈ Q with y 6= 0, we have (x, y) = eDM−yD with D = ( −1 d ), d = (1 − x)y . hζGi −1 In particular, (0, 1) ∈ e , and given (x, 0) ∈ Q, we obtain (x, 0) = (0, 1)EMxE with 1 x−1 E = 0 1 . We have proved that hζGi = 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-element 2 subsets of the projective plane P (Fq). This is a transitive action, because the natural action 2 of G on P (Fq) is 2-transitive. Pick a two-element subset e = {e1, e2} arbitrarily, and consider matrices with respect to the basis (e1, e2, e3), with an arbitrary completion by e3. Clearly, Ge = {Ma,b,Na,b : a, b ∈ Fq}, where 1 0 0 0 1 0 Ma,b = 0 1 0 ,Na,b = 1 0 0 . a b 1 a b −1

A quick calculation shows that ζ = Ma,−a ∈ Z(Ge) for every a ∈ Fq. Since G is a simple group, we obtain for free that the normal subgroup hζGi is equal to G (unless a = 0). Thus Q(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 of G Ge ' Z3 we get hζ i = 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 hζGi = 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 of G Ge ' Z5 we get hζ i = 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 of G Ge ' Z3 we get hζ i = 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 not obvious. For example, a general construction of connected quandles of size 3n was presented by Clark et al. [5], inspired by Galkin [13], by extending the affine quandle Aff(Z3, −1) by a pointed abelian group.

µ Example 6.9. Let A be an abelian group and c ∈ A. We define µ, τ : Z3 → A by 0 = 2, µ µ τ τ τ 1 = 2 = −1 and 0 = 1 = 0, 2 = c, and we define a binary operation on Z3 × A by (x, a) ◦ (y, b) = (−x − y, −a + (x − y)µb + (x − y)τ c). 17 Then G(A, c) = (Z3 × A, ◦) is a connected quandle, called the Galkin quandle corresponding to the pointed group (A, c). It is affine iff 3A = 0. It is latin iff |A| is odd. Two Galkin quandles are isomorphic iff the corresponding pointed groups are isomorphic. See [5] for details.

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 10 S5 6.2 simple 12 S4 6.4 12 A5 6.8 simple 12 A4 o Z4 Gal(A4, 1, (1, 2, 3, 4)) 2 12 (Z3 o Q8) o Z3 2 12 (Z4 o Z3) o Z2 G(Z4, 0) 2 12 (Z4 o Z3) o Z2 G(Z4, 1) 2 12 (Z4 o Z3) o Z2 G(Z4, 2) 4 2 12 (Z2 o Z3) o Z2 G(Z2, (0, 0)) 4 2 12 (Z2 o Z3) o Z2 G(Z2, (1, 1)) 2 15 (Z5 o Z3) o Z2 G(Z5, 0) latin 2 15 (Z5 o Z3) o Z2 G(Z5, 1) latin 15 S6 6.2 simple 15 SL2(4) 6.6 simple . . 2 21 (Z7 o Z3) o Z2 G(Z7, 0) latin 2 21 (Z7 o Z3) o Z2 G(Z7, 1) latin 21 S7 6.2 simple 21 PSL3(2) 6.7 simple . . 2 33 (Z11 o Z3) o Z2 G(Z11, 0) latin 2 33 (Z11 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}. In the column labeled “construction” we either give a reference to a numbered example which uniquely determines the quandle, or we specify how the quandle can be constructed as Gal(G, H, f) of Construction 3.1, or we specify how the quandle can be constructed as G(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 quandles of 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 and Guralnick [11] that connected quandles of order p are affine: the first proof uses Joyce’s characterization of RMlt(Q) for simple quandles, the second proof requires Galois’ result on solvable primitive groups. However, all three proofs still rely on a deep result on conjugacy classes of prime power order by Kazarin [25]. We then mention the result of Graña[16] that connected quandles of order p2 are affine. We conclude with a new, purely group-theoretical proof (modulo Theorem 4.3) of the recent result of McCarron [33] that there are no connected quandles 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 same cardinality. Proof. It is clear that ∼ is an equivalence relation. Let [a], [c] be two equivalence classes of ∼. Since Q is connected, there is θ ∈ RMlt(Q) such that aθ = c. If a ∼ b then θ θ θ θ Rc = Raθ = Ra = Rb = Rbθ , thus c ∼ b , showing that [a] ⊆ [c]. Since θ is one-to-one, we −1 deduce |[a]| ≤ |[c]|. The mapping θ ∈ RMlt(Q) gives the other inequality. Proposition 7.2. Let Q be a connected quandle of prime power order. Then RMlt(Q) is a solvable 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 hxGi is solvable. Let Q be a connected quandle of prime power order, let G = RMlt(Q), and let ζ = Re α α for any e ∈ Q. For every α ∈ G, we have ζ = Re = Reα . For x ∈ Q, taking α ∈ G α α G G such that e = x, we obtain ζ = Rx. Hence ζ = {Rx : x ∈ Q} and thus hζ i = G. By Proposition 7.1, |ζG| is a divisor of |Q|, hence a prime power. Kazarin’s result then implies G that hζ i = 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 invariant with respect to the action of G. An observation by Joyce [23, Proposition 3] says that if Q is simple then G0 is the smallest normal subgroup in G. Since G is solvable, we then must have G00 = 1, hence G0 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 of the affine group over a finite field. Theorem 5.3 now concludes the proof. Grañaproved: 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 a n connected quandle of order 2 . 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 n 2 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. 0 Suppose that Q is a connected quandle of order 2p. Then G = RMlt(Q) ≤ S2p, G acts G transitively on Q by Proposition 2.3, and hζ i = G for some ζ ∈ Z(Ge) by Theorem 4.3, G so, in particular, hZ(Ge) i = G. Theorem 7.5 therefore follows from the group-theoretical Theorem 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 both of the following conditions: (A) G0 is transitive on {1,..., 2p}. G (B) hZ(G1) i = G. We start with two general results on the center of the stabilizer of almost simple primitive groups of degree p and 2p. Both proofs are based on the explicit classiﬁcation of almost simple 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 transitive group 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 ) = h1i. 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. Then K = 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 ) = h1i.

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 elements of U can be represented by pairs [field automorphism, matrix] of the form a 0 τ, BA 20 ∗ d−1 with a ∈ Fq, B ∈ Fq and A ∈ GLd−1(q) and τ ∈ hσi. Two such elements multiply as a1 0 a2 0 a1a2 0 τ1, · τ2, = τ1τ2, τ2 τ2 τ2 B1 A1 B2 A2 B1 + A1 B2 A1 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 C W of index 2, so it has a smaller A-part. (If it had a smaller B-part, this would have to be a submodule for the natural SLd−1(q)-module which is irreducible.) The A-part cannot be smaller if d − 1 ≥ 3, or if d − 1 = 2 and q ≥ 4. In the remaining cases (d − 1 = 2 and q ∈ {2, 3}; respectively d − 1 = 1) the A-part can be smaller by index 2. However we note by inspection that there is no B-part that is fixed by all A-parts by multiplication. We now consider a pair of elements, the second being in V and the first being in CU (V ). τ2 τ2 τ1 τ1 By the multiplication formula the elements commute only if B1 + A1 B2 = B2 + A2 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 moves B1 (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 vector moved by τ1 and violate the equation. This shows that the only element of U commuting with 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 ) = h1i. Proof. As subgroups of index 2 are normal we know that there exists a subgroup V ≤ W as speciﬁed in Lemma 8.4. But then by this lemma

Z(W ) ≤ CW (V ) ≤ CAut(PSLd(q))subspace (V ) = h1i.

Proof of Theorem 8.2. For case (i) of Lemma 8.3, we have that U ∈ {Sp−1, Ap−1} and so also V ∈ {Sp−1, Ap−1}, thus (as p ≥ 5) clearly Z(V ) = h1i. For case (ii) we get from Corollary 8.5 that Z(V ) = h1i. 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 0) establishes the result. 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) = h1i. 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 one of the following groups:

(i) K = A2p, (ii) p = 5, K = A5 acting on 2-sets, 2a (iii)2 p = q + 1, q = r 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 thus G1 ∈ {S2p−1,A2p−1} for which the statement is clearly true. Case (ii) is irrelevant here as p = 5. Case (iii) follows from Corollary 8.5. Case (iv) is again done with an explicit calculation 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 these 0 blocks. 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 by B, let 1 ∈ B ∈ B. Labeling points suitably, we can assume that B = {1, 2}. Let S = G1 be a 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) p and note that M ≤ C2 is either trivial or has exactly p orbits of length 2. Lemma 8.10. If M 6= h1i then T = MS. Proof. If M 6= h1i, then M has orbits of length 2. Consider t ∈ T . If 1t 6= 1 then 1t = 2 is t m −1 in the same M-orbit. Thus there exists m ∈ M such that 1 = 1 , thus tm ∈ S. As p is a prime, H is a primitive group. By the O’Nan-Scott theorem [29], we know that H is either of aﬃne type or almost simple. Lemma 8.11. If H is almost simple, then condition (B) is violated.

ϕ ϕ ϕ Proof. If M 6= h1i then by Lemma 8.10 S = T = H1. But then Z(S) ≤ Z(H1) = h1i by G Theorem 8.6. Thus Z(S) ≤ Ker(ϕ) C G and hZ(S) i= 6 G. If M = h1i 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 that G Z(S) = h1i and thus hZ(S) i= 6 G. ∗ It remains to consider the aﬃne case, i.e. H ≤ Fp o Fp. We can label the p points on which H acts as 0, . . . , p − 1, then the action of the Fp-part is by addition, and that of the ∗ ϕ Fp-part by multiplication modulo p. Without loss of generality assume that T = H1. We may also assume that H is not cyclic as otherwise H0 = h1i and thus G0 ≤ 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 is no 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 with a ∈ {0, 1, 2, 3} and b(ap + 1) = 4(p − 1). If a 6= 0 this implies that b ∈ {1, 2, 3, 4}. Trying out 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 as orbits 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= h1i, thus by Lemma 8.10 we have ϕ ∗ ϕ that S = H1 ≤ Fp. Thus there exists b ∈ S such that H1 = hb i. Lemma 8.13. S = hbi · L.

ϕ ϕ ϕ x Proof. Clearly S ≥ hbi · L. Consider s ∈ S. Then s ∈ H1, thus s = (b ) for a suitable x −x and thus sb ∈ Ker(ϕ) ∩ S = L. ∗ We shall need a technical lemma about ﬁnite ﬁelds. For β ∈ Fp, a subset I ⊂ Fp is called β-closed if Iβ = I, that is x ∈ I ⇔ xβ ∈ I.

∗ ∗ Lemma 8.14. Let α, β ∈ Fp, β 6= 1 and assume that ∅ 6= I ⊂ Fp is β-closed. Then I − 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 a ﬁnite 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 be closed under addition of α(1 − β) 6= 0. But the additive order of a nonzero element in Fp is p, 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. For x this assume to the contrary that b · l ∈ CS(L) with l ∈ L and x a suitable exponent such x x ∗ that b 6∈ L. As L ≤ M is abelian this implies that b ∈ CS(L). Let β ∈ Fp ≤ H be such that (bx)ϕ = β. As bx 6∈ L we know that β 6= 1. p When we consider the conjugation action of G on M ≤ C2 , note that an element of M is determined uniquely by its support (that is the blocks in B whose points are moved by the element), which we consider as a subset of Fp, which is the domain on which H acts. An element g ∈ G acts by conjugation on M with the eﬀect of moving the support of elements in ϕ x the same way as g moves the points Fp. For b to centralize an element a ∈ L, the support I 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 whose ∗ ∗ support I is a proper nonempty subset of Fp. Thus there exists α ∈ Fp, α 6∈ I. That means that if we conjugate a with −α ∈ Fp, the resulting elementa ˜ has support I − α. By assumption 0 6∈ I − α, soa ˜ ∈ L. But by Lemma 8.14 we know that I − α is not x β-closed, that isa ˜ ∈ L is not centralized by b . Corollary 8.16. If H is of aﬃne type, then at least one of conditions (A), (B) is violated.

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(Hulpke) Department of Mathematics, Colorado State University, 1874 Campus Delivery, Ft. Collins, Colorado 80523, U.S.A.

(Stanovsk´y) Department of Algebra, Faculty of Mathematics and Physics, Charles Uni- versity, Sokolovska´ 83, Praha 8, 18675, Czech Republic

(Stanovský,Vojtˇechovský) Department of Mathematics, University of Denver, 2280 S Vine St, Denver, Colorado 80208, U.S.A. E-mail address, Hulpke: [email protected] E-mail address, Stanovský: [email protected] E-mail address, Vojtˇechovský: [email protected]