Quandles with orbit series conditions
Marco Bonatto, Alissa S. Crans, Timur Nasybullov, Glen T. Whitney
aa r X i v : . [ m a t h . G R ] A p r QUANDLES WITH ORBIT SERIES CONDITIONS
M. BONATTO, A. CRANS, T. NASYBULLOV, AND G. WHITNEY
Abstract.
We introduce the notion of an orbit series in a quandle. Using this notion we definefour families of quandles based on finiteness conditions on their orbit series. Intuitively, the classes t OS and t OS n correspond to finitary compositions of trivial quandles while the classes OS and OS n correspond to finitary compositions of connected quandles. We study properties of thesefour families of quandles and explore their relationships with several previously studied families ofquandles: reductive, n -reductive, locally reductive, n -locally reductive, and solvable quandles. Keywords: quandle, orbit series condition, finite type condition, Engel group, nilpotent group.Mathematics Subject Classification: 20N02, 16T25, 57M27. Introduction
A quandle is an algebraic structure whose axioms are derived from the Reidemeister moves onoriented link diagrams. They were first introduced by Joyce [33] and Matveev [38] as an invariantfor knots in R . More precisely, to each oriented diagram D K of an oriented knot K in R onecan associate the quandle Q ( K ) which does not change if we apply the Reidemeister moves to thediagram D K . The knot quandle Q ( K ) can be constructed as the quotient of the free quandle F Q n on the n arcs of D K by relations which can be obtained from the crossings of D K . Joyce andMatveev proved that two knot quandles Q ( K ) and Q ( K ) are isomorphic if and only if there isa homeomorphism of the ambient R which maps K to K and which respects their orientations.Over the years, quandles have been investigated by various authors in order to construct newinvariants for knots and links (see, for example, [18, 34, 39]).The knot quandle is a very strong invariant for knots in R , however, usually it is very difficultto determine if two knot quandles are isomorphic. Sometimes homomorphisms from knot quandlesto simpler quandles provide useful information which helps understand whether two knot quandlesare isomorphic. This leads to the necessity of studying broader classes of quandles from the alge-braic point of view. Algebraic properties of quandles including their automorphisms and residualproperties have been investigated, for example, in [2, 4, 9, 13, 21, 30, 31, 40, 42]. A (co)homologytheory for quandles and racks has been developed in [19, 27, 28, 41] that led to stronger invariantsfor knots and links. Many new constructions of quandles have been introduced, for example, in[5, 6, 15, 22]Quandles also find applications in the study of the set-theoretical Yang-Baxter equation. TheYang-Baxter equation first appeared in theoretical physics and statistical mechanics in the works ofYang [45] and Baxter [10, 11]. The notion of a set-theoretical solution of the Yang-Baxter equationwas introduced by V. Drinfel’d in the context of quantum groups (see [23]) as a pair ( X, r ) , where X is a set and r ∶ X × X → X × X is a bijective map such that ( r × id )( id × r )( r × id ) = ( id × r )( r × id )( id × r ) . If Q is a quandle with operation ⊳ , then the pair ( Q, r ) where r ∶ Q × Q → Q × Q is the map givenby r ( x, y ) = ( x ⊳ y, x ) for x, y ∈ Q provides such a solution. he recent paper [7] constructed a general method whereby a given solution of the set-theoreticalYang-Baxter equation on an arbitrary algebraic system X can be used for constructing a repre-sentation of the virtual braid group V B n by automorphisms of the algebraic system X . Also [8]introduced a method to construct an invariant for virtual knots and links from a given solution ofthe set-theoretical Yang-Baxter equation on an algebraic system X . When X is a quandle with thecorresponding Yang-Baxter solution as above then using procedures described in [7, 8] it is possibleto construct a representation V B n → Aut ( X ) and a quandle invariant for virtual links. However, inorder to apply these procedures to X and r , the quandle X must satisfy certain specific conditionsdescribed in [7, 8]. In this regard, there is a natural problem of constructing specific families ofquandles that are convenient to work with.Turning to group theory for inspiration, convenient families of groups include those satisfying cer-tain finite type conditions: (locally/residually) finite groups, (locally/residually) nilpotent groups,(locally/residually) solvable groups, etc. This naturally leads to the idea of introducing similarfamilies of quandles. This idea was partially implemented in [16] using commutator theory fromuniversal algebra to introduce and study nilpotent and solvable quandles. Other previously studiedfamilies of quandles given by finite type conditions are reductive quandles (see [17], and [32] wherethey are called ‘multipermutational’) and locally reductive quandles (see [43], where this notion isconsidered but not specifically named).Here we introduce the notion of an orbit series in a quandle. Using this notion we define fourfamilies of quandles based on finiteness conditions on their orbit series. Intuitively, the classes t OS and t OS n correspond to finitary compositions of trivial quandles while the classes OS and OS n correspond to finitary compositions of connected quandles. In particular, the class of quasi-trivial quandles discussed in [25] corresponds exactly to t OS here. We study properties of thesefour families of quandles and explore their relationships with several previously studied families ofquandles: reductive, n -reductive, locally reductive, n -locally reductive, and solvable quandles.The paper is organized as follows. In Section 2 we recall the necessary preliminaries aboutquandles. In Section 3 we discuss the notions of reductive and locally reductive quandles andderive some of their properties. In Section 4 we introduce the notion of an orbit series in a quandle,give definitions of the families OS , OS n , t OS , t OS n and provide several examples. Section 5is devoted to the study of properties of the families OS , OS n , t OS , t OS n and of the quandlesfrom these families: in Section 5.1 we study the behavior of the families OS , OS n , t OS , t OS n under taking subquandles, homomorphic images, and direct products, in Section 5.2 we study thesubquandle structure of quandles from the families OS , OS n , t OS , t OS n , and in Section 5.3 westudy connections between the families t OS , t OS n and the families of reductive, locally reductive,and solvable quandles. Acknowledgements.
T. Nasybullov is supported by the Ministry of Science and Higher Educationof the Russian Federation, government program of Sobolev Institute of Mathematics SB RAS,project 0250-2019-0001. Alissa S. Crans is supported by a grant from the Simons Foundation(
Preliminaries
Quandles.
A quandle Q is a set equipped with a binary operation ⊳ satisfying the followingthree axioms:(1) a ⊳ a = a for all a ∈ Q ,(2) the map L a ∶ b ↦ a ⊳ b is a bijection of Q for all a ∈ Q ,(3) a ⊳ ( b ⊳ c ) = ( a ⊳ b ) ⊳ ( a ⊳ c ) for all a, b, c ∈ Q .Axioms (2) and (3) are equivalent to the fact that the map L x ∶ Q → Q is an automorphismof Q for all x ∈ Q . The group Inn ( Q ) = ⟨{ L x ∶ x ∈ Q }⟩ generated by all L x is called the inner utomorphisms of Q . The group Trans ( Q ) = ⟨{ L x L − y ∶ x, y ∈ Q }⟩ is a normal subgroup in Inn ( Q ) called the transvection group of Q (sometimes called the displacement group of Q , Dis ( Q ) ). Bothgroups Inn ( Q ) and Trans ( Q ) act on Q . The orbit of an element x ∈ Q under the action of Inn ( Q ) coincides with the orbit of x under the action of Trans ( Q ) . This orbit is denoted by Orb ( x, Q ) andis called the orbit of x in Q , or the component of x in Q . If Q = Orb ( x, Q ) for some x ∈ Q , then Q is said to be connected.The following are well-known examples of quandles. ● The simplest example of a quandle is the trivial quandle on a set X : Q = ( X, ⊳ ) where x ⊳ y = y for all x, y ∈ X . If ∣ X ∣ = n , then the trivial quandle on X is denoted by T n . ● Let G be a group. For elements x, y ∈ G denote the conjugate of y by x as y x = x − yx . Foran arbitrary integer n , the set G with the operation x ⊳ y = y x n = x − n yx n defines a quandle called the n -th conjugation quandle of the group G , denoted by Conj n ( G ) .For the sake of simplicity we will use the symbol Conj ( G ) for the first conjugation quandleConj ( G ) . Note that a quandle Conj ( G ) is trivial. If H ⊂ G is closed under the operationof conjugation, then Conj ( H ) is the quandle on H with the operation given by g ⊳ h = g − hg for g, h ∈ H . ● Let ϕ be an automorphism of a group G . Then the set G with the operation x ⊳ y = ϕ ( yx − ) x forms a quandle called the generalized Alexander quandle of the group G with respect to theautomorphism ϕ , denoted by Alex ( G, ϕ ) . Alexander quandles were studied, for example,in [2, 20, 21] and in [14] under the name of principal quandles. ● If G is an abelian group then the inversion operation − is an automorphism of G , andAlex ( G, −) is known as the Takasaki quandle [44] of G . For C n the cyclic group of order n ,Alex ( C n , −) is called the dihedral quandle on n elements.2.2. Congruences and homomorphisms.
Let α be an equivalence relation on a set Q . We write a α b to mean that a, b ∈ Q are α -related. We denote the class of a with respect to α by [ a ] α (andsometimes drop the subscript α ) and the quotient set by Q / α = {[ a ] α ∶ a ∈ Q } .A congruence of a quandle Q is an equivalence relation α ⊆ Q × Q that respects the algebraicstructure. Namely, the operation on Q / α defined as [ a ] α ⊳ [ b ] α = [ a ⊳ b ] α is well defined and provides a quandle structure on Q / α . Then congruences are the equivalencerelations such that if a α b and c α d , then ( a ⊳ c ) α ( b ⊳ d ) , and if a ⊳ x = c and b ⊳ y = d then x α y .The canonical surjective map a ↦ [ a ] α is a quandle homomorphism. If h ∶ Q → Q ′ is a quandlehomomorphism, then the equivalence relationker ( h ) = {( a, b ) ∈ Q × Q ∶ h ( a ) = h ( b )} is a congruence of Q . By virtue of the second homomorphism theorem for general algebraic struc-tures [12], there exists a one-to-one correspondence between congruences and kernels of homomor-phisms, hence congruences and homomorphic images are essentially the same thing. The con-gruences of a quandle form a lattice denoted by Con ( Q ) whose minimum is the identity relation0 Q = {( a, a ) ∶ a ∈ Q } and maximum is 1 Q = Q × Q . Then Con ( Q / α ) is given by the congruences β / α = {([ a ] α , [ b ] α ) ∈ Q / α × Q / α ∶ a β b } or every β ∈ Con ( Q ) such that α ⊆ β . A class K of quandles is said to be closed under homomorphicimages if Q / α belongs to K for every quandle Q ∈ K and every congruence α ∈ Con ( Q ) . If α is acongruence of Q and a ∈ Q , then the class [ a ] α is clearly a subquandle of Q . A class K of quandlesis said to be closed under extensions if Q ∈ K whenever Q / α ∈ K and [ a ] α ∈ K for every a ∈ Q .Any congruence α of a quandle Q induces a surjective homomorphism of groups π α ∶ Inn ( Q ) → Inn ( Q / α ) defined by L a ↦ L [ a ] α , a ∈ Q which restricts and corestricts to the displacement groups. In particular, for every a ∈ Q and every h ∈ Inn ( Q ) the equation [ h ( a )] α = π α ( h )([ a ] α ) holds. Hence,(1) {[ h ( a )] α ∶ h ∈ N } = { π α ( h )([ a ] α ) ∶ h ∈ N } for every a ∈ Q and every N ≤ Inn ( Q ) . The kernel of π α is denoted byInn α = { h ∈ Inn ( Q ) ∶ h ( a ) α a for every a ∈ Q } , and the kernel of the restriction π α ∣ Trans ( Q ) ∶ Trans ( Q ) → Trans ( Q / α ) is denoted by Trans α . Wecan also define the transvection group relative to α asTrans α = ⟨{ L a L − b ∶ a α b }⟩ . The set of all normal subgroups of the group Inn ( Q ) that belong to Trans ( Q ) is denoted byNorm ( Q ) = { N ≤ Trans ( Q ) ∶ N ⊴ Inn ( Q )} . In particular, Trans α and Trans α belong to Norm ( Q ) .If N belongs to Norm ( Q ) , then we define the following congruences of Q as in [16] by: O N = {( a, b ) ∈ Q × Q ∶ b = n ( a ) , for some n ∈ N } , i. e., O N is the orbit decomposition with respect to the action of N . The O N decomposition willplay a significant role in Section 3.1.From the third quandle axiom it follows that, for L x , L y ∈ Inn ( Q ) , L x ⊳ y = L x L y L − x and thus the map L ∶ x ↦ L x is a quandle homomorphism from Q to Conj − ( Inn ( Q )) , with imageits subquandle L ( Q ) = { L x ∶ x ∈ Q } . The kernel of the homomorphism L is a congruence denotedby λ Q , i.e., x λ Q y if and only if L x = L y . If L is injective then Q is said to be faithful . Note thatTrans α = α ⊆ λ Q . For any quandle we have the following inductively defined chainof homomorphic images: L ( Q ) = Q, L n + ( Q ) = L ( L n ( Q )) for n ∈ N . We define the enveloping group of a quandle Q in terms of generators and relations as G Q = ⟨ e x ∣ e x e y e − x = e x ⊳ y , x, y ∈ Q ⟩ . The canonical map i ∶ x ↦ e x is a quandle homomorphism from Q to Conj ( G Q ) with the image E ( Q ) = { e x ∶ x ∈ Q } . The map G Q → Inn ( Q ) given by e x ↦ L x on the generators can be extendedto a surjective group homomorphism with central kernel [24, Proposition 2.37].2.3. Abelian, nilpotent and solvable quandles.
The properties of abelianess, nilpotence andsolvability for quandles, in the sense of commutator theory [26], have been investigated in [16].For an algebraic structure A , the commutator is a binary operation on the congruence lattice,defined using the centralization as follows. Let α , β , and δ be congruences of A . We say that α centralizes β over δ if for every ( n + ) -ary term operation t (i.e., any function t ∶ A n + Ð→ A obtained by an expression in the basic operations), every pair a α b , and every u β v , . . . , u n β v n we have t ( a, u , . . . , u n ) δ t ( a, v , . . . , v n ) implies t ( b, u , . . . , u n ) δ t ( b, v , . . . , v n ) . he commutator of α , β , denoted by [ α, β ] , is the smallest congruence δ such that this implicationholds. A congruence α is called abelian if [ α, α ] = A and central if [ α, A ] = A .An algebraic structure A is called abelian if the congruence 1 A is abelian. It is called nilpotent (resp. solvable ) if and only if there is a chain of congruences0 A = α ≤ α ≤ . . . ≤ α n = A such that α i + / α i is a central (resp. abelian) congruence of A / α i for all i ∈ { , , . . . , n − } . Thelength of the smallest such series is called the length of nilpotence (resp. solvability).As for groups, one can define the series γ ( A ) = A , γ i + ( A ) = [ γ i ( A ) , A ] , and γ ( A ) = A , γ i + ( A ) = [ γ i ( A ) , γ i ( A )] , and prove that an algebraic structure A is nilpotent (resp. solvable) of length n if and only if γ n ( A ) = A (resp. γ n ( A ) = A ).In the case of quandles, these properties turn out to be completely determined by group theoreticproperties of the transvection group. Recall that a group acting on a set is semiregular if thepointwise stabilizers are trivial. Theorem 2.1. [16]
A quandle Q is solvable (resp. nilpotent) if and only if Trans ( Q ) is solvable(resp. nilpotent). In particular, Q is abelian if and only if Trans ( Q ) is abelian and semiregular. A quandle Q is is said to be medial if ( a ⊳ b ) ⊳ ( c ⊳ d ) = ( a ⊳ c ) ⊳ ( b ⊳ d ) for every a, b, c, d ∈ Q . Note that Joyce calls such quandles “abelian” [33]. We choose to use theword “medial” (as in [31]) to avoid the conflict with the term “abelian” as in used Theorem 2.1above. A quandle is medial if and only if its transvection group is abelian. Therefore all abelianquandles are medial; however, due to Theorem 2.1 the converse is false in general.3. Reductive and locally reductive quandles n-Reductive quandles.
Let n be a positive integer. A quandle Q is said to be n-reductive if either of the following two equivalent identities holds: ( . . . (( a ⊳ c ) ⊳ c ) . . . ) ⊳ c n − ) ⊳ c n = ( . . . (( b ⊳ c ) ⊳ c ) . . . ) ⊳ c n − ) ⊳ c n (2) ( . . . (( a ⊳ c ) ⊳ c ) . . . ) ⊳ c n − ) ⊳ c n = ( . . . ( c ⊳ c ) . . . ) ⊳ c n − ) ⊳ c n for all a, b, c , c , . . . , c n ∈ Q . Note that every 1-reductive quandle is trivial.Let the set of all n -reductive quandles be denoted by R n ( n ≥ ) and let R ω = ⋃ n ≥ R n . Theproperty of being reductive (i.e., belonging to R ω ) can be characterized in terms of the followinginductively defined descending chain of congruences O Q = Q , O n + Q = O Trans O nQ (3)for n ∈ N . It is easy to show that Trans O nQ / α = π α ( Trans O nQ ) for every n ∈ N . We drop the argument Q when it is clear from the context. The proposition following the next lemma provides equivalentdefinitions of reductive quandles. Lemma 3.1.
Let Q be a quandle and α ∈ Con ( Q ) . Then [ a ] O nQ / α = {[ b ] α ∶ b O nQ a } . roof. According to (1) we have to show that π α ( Trans O nQ ) = Trans O nQ / α . If n =
0, thenTrans O Q / α = Trans ( Q / α ) = π α ( Trans ( Q )) = π α ( Trans O Q ) . The generators of Trans O n + Q / α are of the form L a L − b where b = g ( a ) for some g ∈ Trans O nQ / α . Henceby induction, g = π α ( h ) for some h ∈ Trans O nQ and so L [ a ] L − [ b ] = L [ a ] L − π α ( h )([ a ]) = π α ( L a L − h ( a ) ) . Thus, L [ a ] L − [ b ] ∈ π α ( Trans O n + Q / α ) . (cid:3) Proposition 3.2.
Let Q be a quandle. Then the following are equivalent: (1) Q ∈ R n , (2) ∣ L n ( Q )∣ = , (3) Inn ( Q ) is nilpotent of nilpotency class n − , (4) O nQ = Q .In particular, the only faithful reductive quandle is the trivial quandle with one element.Proof. The equivalence between (1), (2) and (3) is Theorem 4.1 of [32]. We will show (1) isequivalent to (4) using induction on n .(1) ⇒ (4). The basis of induction ( n =
1) is simple: in this situation Trans ( Q ) =
1, and therefore O Q = Q . For the inductive step let Q be an n -reductive quandle for n >
1. Then Q / λ Q = L ( Q ) is ( n − ) -reductive and by induction O n − L ( Q ) = L ( Q ) . Thus, by Lemma 3.1, [ a ] O n − Q ⊆ [ a ] λ Q for every a ∈ Q , i.e., O n − Q ⊆ λ Q and so Trans O n − Q ≤ Trans λ Q =
1. Hence O nQ = Q and so (1) implies (4).(4) ⇒ (1). If n =
1, then O Q = Q means that Q is trivial, and therefore Q is 1-reductive. Let O n + Q = Q and Q ′ = Q / O nQ . The group homomorphism π O nQ maps Trans O kQ onto Trans O kQ /O nQ andthen by Lemma 3.1, O kQ ′ = O kQ / O nQ for every k ≤ n . Therefore Q ′ has a chain of congruences1 Q ′ = O Q ′ ⊆ O Q ′ ⊆ . . . ⊆ O n − Q ′ ⊆ O nQ ′ = O nQ / O nQ = Q ′ . By induction, Q ′ is n -reductive. Since O n + Q = Q , the orbits of Trans O nQ are trivial, i. e., Trans O nQ = O nQ ⊆ λ Q . Thus, since Q / O nQ is n -reductive and O nQ ⊆ λ Q , then by [17], Q is ( n + ) -reductive. (cid:3) Corollary 3.3.
Let G be a group. Then Conj ( G ) ∈ R n if and only if G is nilpotent of class n .Proof. The statement follows from Proposition 3.2 due to the equality Inn ( Conj ( G )) = G / Z ( G ) . (cid:3) Locally reductive quandles.
Let n be a positive integer. A quandle Q is said to be n -locallyreductive if the equality(4) ( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n = b holds for all elements a, b ∈ Q . It is clear that every 1-locally reductive quandle is trivial. Wedenote the class of all n -locally reductive quandles by LR n ( n ≥ ) and let LR ω = ⋃ n ≥ LR n . Every n -reductive quandle is obviously n -locally reductive, therefore we have the inclusions(5) LR n − ⊂ LR n ⊂ LR ω ∪ ∪ ∪ R n − ⊂ R n ⊂ R ω If Q is a medial quandle, then (2) and (4) are equivalent [31]. That is, Q ∈ R n if and only if Q ∈ LR n . We want to understand connections between the classes R ω and LR ω in general. emma 3.4. The class LR ω is closed under taking subquandles, homomorphic images, finite directproducts and extensions. Moreover, if Q / α ∈ LR n and [ a ] α ∈ LR m for all a ∈ Q then Q ∈ LR n + m .Proof. Each class LR n is a variety, so, clearly LR ω is closed under taking subquandles, homomor-phic images and finite direct products since the classes { LR n ∶ n ∈ N } form a chain. Let Q / α ∈ LR n and [ a ] α ∈ LR m for every a ∈ Q . Then [( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n ] α = [ b ] α and so ( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ m = b for every a, b ∈ Q . (cid:3) Corollary 3.5.
Let Q be a quandle. If all the orbits of Q are n -locally reductive, then Q is ( n + ) -locally reductive.Proof. The factor Q / O Q is 1-locally reductive. Hence we can apply Lemma 3.4. (cid:3) Let G be a group and let a, b ∈ G . Following standard notation, we use [ b, k a ] to denote thefollowing inductively defined element of G : [ b, a ] = [ b, a ] , [ b, n + a ] = [ b, [ b, n a ]] for n ∈ N . A subset H of a group G is called an n -Engel subset if [ b, n a ] = a, b ∈ H . Anelement g ∈ G is an n -Engel element of G if [ b, n g ] = b ∈ G . Proposition 3.6.
Let G be a group, and H ⊆ G be a subset of G closed under conjugation. Thenthe following statements are equivalent. (1) Conj ( H ) is n -locally reductive. (2) H is an n -Engel subset of G .In particular, Conj ( G ) is n -locally reductive if and only if G is an n -Engel group.Proof. Using induction on n we will show that for a, b ∈ G ( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n = [ b, n − a ] − b [ b, n − a ] for every n ≥
2. The basis of induction ( n =
2) follows from the equality ( a ⊳ b ) ⊳ b = a − b − aba − ba = [ b, a ] − b [ b, a ] . The induction step follows from the equalities ( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n + = ( . . . ((( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b )´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ n ⊳ b = [ b, n − a ] − b − [ b, n − a ] bbb − [ b, n − a ] − b [ b, n − a ] = [ b, n a ] − b [ b, n a ] . Hence, H is n -locally reductive if and only if b = [ b, n − a ] − b [ b, n − a ] , or, equivalently [ b, n a ] = a, b ∈ H , i. e., if H is an n -Engel subset. (cid:3) Proposition 3.6 describes the necessary and sufficient condition under which Conj ( H ) is n -locallyreductive and together with Corollary 3.3 shows that the inclusions in (5) are all strict. Thefollowing proposition states that this is the only case that must be considered, i. e., the problem ofdetermining if a quandle is locally reductive can be reduced to the class of conjugation quandles. roposition 3.7. Let Q be a quandle. Then the following statements are equivalent. (1) Q ∈ LR ω . (2) E ( Q ) is an n -Engel subset of G Q for some n ∈ N . (3) L ( Q ) is an n -Engel subset of Inn ( Q ) for some n ∈ N .In particular, if L ( Q ) is n -locally reductive, then Q is ( n + ) -locally reductive.Proof. (1) ⇒ (2) ⇒ (3) Let Q be a quandle and E ( Q ) = { e x ∶ x ∈ Q } . Then the mappings Qx →↦ E ( Q ) e x →↦ L ( Q ) L x are quandle homomorphisms. Hence if Q (resp. E ( Q ) ) is a n -locally reductive quandle then E ( Q ) (resp. L ( Q ) ) is a n -locally reductive conjugation quandle. Therefore, by Proposition 3.6, E ( Q ) (resp. L ( Q ) ) is an n -Engel subset of G Q (resp. Inn ( Q ) ).(3) ⇒ (1) The quandle L ( Q ) = Q / λ Q is a n -locally reductive conjugation quandle and the blocksof λ Q are 1-locally reductive. Thus by Lemma 3.4 the quandle Q is ( n + ) -locally reductive. (cid:3) Corollary 3.8.
Let Q be a quandle. If Inn ( Q ) is n -Engel then Q is ( n + ) -locally reductive.Proof. The quandle L ( Q ) = Q / λ Q embeds into the n -locally reductive conjugation quandle Conj ( Inn ( Q )) and therefore it is n -locally reductive. Then we can apply Proposition 3.7. (cid:3) Let G be a finite group. Then G is n -Engel for some n if and only if G is nilpotent [47]. So,Conj ( G ) ∈ R ω if and only if Conj ( G ) ∈ LR ω . We show that if Q is a finite quandle (not necessarilya conjugation quandle), then Q ∈ R ω if and only if Q ∈ LR ω .A subset H of a group G is called normal if x − Hx = H for all x ∈ G . The Fitting subgroup ofa group G is the subgroup generated by all nilpotent normal subgroups of G . The following resultfollows from [1, Theorem 2.8]. Theorem 3.9.
Let G be a group which satisfies the maximal condition on abelian subgroups. Thenevery normal n -Engel subset of G for n ≥ is contained in the Fitting subgroup of G . This theorem implies the following statement.
Theorem 3.10.
Let Q be a finite quandle. Then Q is reductive if and only if Q is locally reductive.In particular, if Q is locally reductive connected or faithful then ∣ Q ∣ = .Proof. If Q is reductive, then it is obviously locally reductive (since R ω ⊂ LR ω ). Suppose that Q is finite n -locally reductive quandle. According to Proposition 3.7, the group Inn ( Q ) is generatedby the normal n -Engel subset { L a ∶ a ∈ Q } , and since Inn ( Q ) is finite, it satisfies the maximalcondition on abelian subgroups. From Theorem 3.9, it follows that Inn ( Q ) coincides with itsFitting subgroup. Since Inn ( Q ) is finite, its Fitting subgroup is nilpotent, and therefore Inn ( Q ) isnilpotent. Thus, by Proposition 3.2, Q is reductive. (cid:3) For medial quandles, reductivity and local reductivity are equivalent [31], so, medial locallyreductive quandles are not connected. Hence, if a locally reductive quandle has a medial connectedfactor, then it has cardinality one. From this observation we can provide the following propositionabout the transvection group of connected locally reductive quandles.
Proposition 3.11.
Let Q be a connected n -locally reductive quandle. Then Trans ( Q ) is a perfectgroup, i.e., Trans ( Q ) = [ Trans ( Q ) , Trans ( Q )] , and Q has no finite factors.Proof. According to [13] the factor of Q with respect to the congruence α = O [ Trans ( Q ) , Trans ( Q )] (given by the orbits of the group [ Trans ( Q ) , Trans ( Q )] ) is a connected medial locally reduc-tive quandle. It follows from Theorem 3.10 that ∣ Q / α ∣ =
1, therefore Trans α = Trans ( Q ) ≤ [ Trans ( Q ) , Trans ( Q )] and Trans ( Q ) = [ Trans ( Q ) , Trans ( Q )] .Finite locally reductive quandles are not connected, so Q has no finite factors. (cid:3) . Orbit series in quandles
Definitions and examples.
Let Q = Q be a quandle and x ∈ Q . Let Q = Orb ( x , Q ) andchoose an element x ∈ Q . Continuing in the same way (choosing x i + ∈ Q i + = Orb ( x i , Q i ) , letting Q i + = Orb ( x i + , Q i + ) , and so on) we obtain a sequence of subquandles:(6) Q = Q ≥ Q ≥ Q ≥ Q ≥ . . . Such a series of subquandles is called the orbit series defined by the elements { x i ∶ i ∈ N } . Notethat the same orbit series can be obtained by different sequences of elements. If an orbit series canbe obtained by a constant sequence of elements, namely all x i = x for some x ∈ Q , then we say thatthis orbit series is the principal orbit series of x . A quandle Q has a unique orbit series if and onlyif it is connected. We will use the following terminology: ● A quandle Q satisfies the descending orbit series condition if each orbit series stabilizes,i. e., for each orbit series as in (6) there exists k ∈ N such that Q k = Q k + . We denote by OS the class of quandles which satisfy this condition. ● A quandle Q satisfies the n -bounded descending orbit series condition if for each orbit seriesas in (6) there exists k ≤ n such that Q k = Q k + . We denote the class of quandles whichsatisfy this condition by OS n and let OS ω = ⋃ ∞ n = OS n . ● A quandle Q satisfies the trivializing orbit series condition if each orbit series as in (6)stabilizes on the trivial quandle with one element, i. e., there exists k ∈ N such that Q k hasonly one element. The class of all quandles which satisfy this condition will be denoted by t OS . ● A quandle Q satisfies the n -bounded trivializing orbit series condition if for each orbit seriesas in (6) there exists k ≤ n such that Q k is a trivial quandle with one element. The class ofquandles which satisfy this condition is called t OS n and t OS ω = ⋃ ∞ n = t OS n .It is clear that every finite quandle satisfies the orbit series condition. The following inclusionsclearly follow from the definitions:(7) t OS ⊂ t OS ⊂ . . . ⊂ t OS ω ⊂ t OS ∩ ∩ ∩ ∩ OS ⊂ OS ⊂ . . . ⊂ OS ω ⊂ OS The following examples show that all inclusions in (7) are strict.
Example 4.1.
The dihedral quandle Alex ( C n , − ) is a union of two orbits each of which is isomor-phic to Alex ( C n , − ) [6, Proposition 3.2]. Therefore for 1 ≤ k ≤ n + k -th member of every orbitseries of Alex ( C n , − ) is isomorphic to Alex ( C n − k + , − ) . This means that Alex ( C n , − ) belongs to t OS n + ∖ t OS n and to OS n + ∖ OS n . So, the inclusions OS n ⊂ OS n + , OS n ⊂ OS ω , t OS n ⊂ t OS n + ,and t OS n ⊂ t OS ω are strict. Example 4.2.
Let Q n = Alex ( C n , − ) ∈ t OS n + ∖ t OS n . Then the disjoint union Q = ⊔ n ≥ Q n belongs to t OS ∖ t OS ω , and so the inclusions OS ω ⊂ OS and t OS ω ⊂ t OS are strict. Example 4.3.
Every connected quandle belongs to OS . If Q is a connected non-trivial quandle,then it has a unique orbit series Q = Q = . . . , which of course does not stabilize on a trivial quandlewith one element. Therefore, the inclusions t OS n ⊂ OS n , t OS ω ⊂ OS ω , and t OS ⊂ OS are strict.The orbit conditions defined above can captured by the properties of the following tree. Let Q be a quandle. Denote by V ( Q ) the following inductively defined set of subquandles of Q .(1) Q ∈ V ( Q ) , and(2) if P ∈ V ( Q ) , and R ≠ P is an orbit in P , then R ∈ V ( Q ) . he graph Γ ( Q ) = ( V ( Q ) , E ( Q )) has vertices V ( Q ) and edges E ( Q ) = {( P, R ) ∶ P, R ∈ V ( Q ) , R is an orbit in P } . It is clear that Γ ( Q ) is a tree with root Q and that every leaf P is a connected subquandle of Q : If P is not connected, then it has an orbit R which does notcoincide with P and then we have an edge ( P, R ) , so P is not a leaf. If the tree has finite depth,then the leaves are the maximal connected subquandles of Q . We call Γ ( Q ) the orbit tree of thequandle Q . Example 4.4.
The following are examples of orbit series trees. ● If Q is a connected quandle, then the graph Γ ( Q ) has only one vertex Q . ● For the trivial quandle T n = { x , . . . , x n } we have: T n ◆◆◆◆◆◆◆◆♥♥♥♥♥♥♥♥♥♥ ✿✿✿✿✿✿⑧⑧⑧⑧⑧ { x } { x } . . . . . . { x n } ● For the dihedral quandle Alex ( C , − ) we have: C ▼▼▼▼▼▼▼qqqqqqq + C ❅❅❅❅❅⑦⑦⑦⑦⑦ + C ❅❅❅❅❅⑦⑦⑦⑦⑦ { } { } { } { } The following statement obviously follows from the definitions of the classes OS , OS n , t OS , and t OS n together with the definition of the orbit series tree. Proposition 4.5.
Let Q be a quandle. Then (1) Q ∈ OS if and only if each branch of Γ ( Q ) has finite depth. (2) Q ∈ OS n if and only if each branch of Γ ( Q ) has depth which is less then or equal to n . (3) Q ∈ t OS if and only if Q ∈ OS and every leaf is a one-element subquandle of Q . (4) Q ∈ t OS n if and only if Q ∈ OS n and every leaf is a one-element subquandle of Q . Intuitively, this means quandles from the families t OS and t OS n are assembled from one-elementtrivial quandles in a finitary way. Similarly, quandles from the families OS and OS n are assembledfrom connected quandles.Let us characterize principal orbit series. Proposition 4.6.
Let Q be a quandle and U = { Q i ∶ i ∈ N } be an orbit series. Then U is theprincipal orbit series of x if and only if x ∈ ⋂ i ∈ N Q i .Proof. Clearly if U is the principal orbit series of x then x ∈ ⋂ i ∈ N Q i . Conversely, let x be anarbitrary element from ⋂ Q i . Since Q i + = Orb ( x i , Q i ) and x ∈ Q i + , the orbit of x i in Q i and theorbit of x in Q i must coincide. So, we have Orb ( x, Q i ) = Orb ( x i , Q i ) = Q i + . (cid:3) If Q satisfies the descending orbit series condition, then each orbit series in Q stabilizes, thereforeeach orbit series has nontrivial intersection and Proposition 4.6 has the following corollary. Corollary 4.7.
Let Q ∈ OS . Then every orbit series in Q is principal. Every finite quandle satisfies the descending orbit series condition, leading to:
Corollary 4.8.
Let Q be a finite quandle. Then every orbit series in Q is principal. here are infinite quandles and orbit series in these quandles which have empty intersection. So,there are quandles with non-principal orbit series; we give a concrete example. Example 4.9.
Let Q = Alex ( Z , − ) be the Takasaki quandle on Z , also known as the infinite dihedralquandle. Specifically, the operation in Q is given by a ⊳ b = a − b for integers a and b . If a ∈ Q ,then the orbit of a in Q has the formOrb ( a, Q ) = { b − a ∣ b ∈ Z } = { c ∣ c and a have the same parity } . Therefore Q has two orbits: Orb ( , Q ) = { even numbers } and Orb ( , Q ) = { odd numbers } . Fromthe equations ( a ) ⊳ ( b ) = ( a − b ) and ( a + ) ⊳ ( b + ) = ( ( a − b ) + ) it follows that each of these two orbits is isomorphic to Q . By the same argument, the orbits withinthese orbits will again be isomorphic to Q . So, then if we construct an orbit series Q = Q ≥ Q ≥ Q ≥ . . . in Q , then each of the quandles Q i will be isomorphic to Q . In fact, by direct computation,the orbit tree of Q is as follows: Z ❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❲❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ + Z ◗◗◗◗◗◗◗◗♠♠♠♠♠♠♠♠ + Z ◗◗◗◗◗◗◗◗♠♠♠♠♠♠♠♠ + Z ❈❈❈❈❈④④④④④ + Z ❈❈❈❈❈④④④④④ + Z ❈❈❈❈❈④④④④④ + Z ❈❈❈❈❈④④④④④ + Z + Z + Z + Z + Z + Z + Z + Z The orbit series of Q are thus in one-to-one correspondence with the 2-adic integers Z , i.e., thesubring of ∏ n ≥ Z / n Z given by the elements ( a n ) n ≥ such that a n = π n + ( a n + ) for every n ≥ π n + is the canonical map making the following diagram commutative Z / / (cid:15) (cid:15) Z / n ZZ / n + Z π n + ttttttttt . The integers embed into Z as the eventually constant sequences. In particular x ∈ Z is in theintersection of the orbit series { a n + n Z ∶ n ∈ N } if and only if, in Z , x = ( a , a , . . . ) . So chooseany 2-adic integer not in Z , say concretely 1 / = ( , , , , , , , . . . ) . The intersection of theorbit series corresponding to 1 / Orbit series and the structure of quandles
Closure properties.
Here we study the behavior of the families OS , OS n , t OS , t OS n undertaking subquandles, homomorphic images, direct products, and extensions. Lemma 5.1.
Let Q be a quandle, α ∈ Con ( Q ) and let { Q n ∶ n ∈ N } and {( Q / α ) n ∶ n ∈ N } be theorbit series determined by the sequences { a n ∶ n ∈ N } and {[ a n ] α ∶ n ∈ N } , respectively. Then ( Q / α ) n = {[ b ] α ∶ b ∈ Q n } for every n ∈ N . roof. We induct on n to show π α ( Inn ( Q n )) = Inn (( Q / α ) n ) . Equation (1) from Section 2.2 is thebase case. Now, assume that π α ( Inn ( Q n )) = Inn (( Q / α ) n ) . Let π α ( h ) ∈ Inn (( Q / α ) n ) , then L π α ( h )([ a n ]) = π α ( L h ( a n ) ) ∈ π α ( Inn ( Q ) n + ) and so Inn (( Q / α ) n + ) = π α ( Inn ( Q n + )) .Using this, we see that ( Q / α ) n = { h ([ a n ]) ∶ h ∈ Inn (( Q / α ) n − )} = { π α ( h )([ a n ]) ∶ h ∈ Inn ( Q n − )} = {[ h ( a n )] ∶ h ∈ Inn ( Q n − )} = {[ b ] α ∶ b ∈ Q n } which completes the proof. (cid:3) Proposition 5.2.
The classes t OS n , t OS ω , t OS , OS n , OS ω , and OS are closed under takinghomomorphic images and finite direct products. The classes t OS n , t OS ω and t OS are also closedunder taking subquandles.Proof. We provide the details for the case of t OS n ; other cases are proved analogously. Let Q ∈ t OS n and α ∈ Con ( Q ) . We will show that Q / α ∈ t OS n . According to Lemma 5.1, the orbit series in Q / α are images of orbits series of Q under the canonical map. Hence, if ∣ Q n ∣ =
1, then ∣( Q / α ) n ∣ = Q ∈ t OS n and P be a subquandle of Q . Let P = P and P i + = Orb ( x i , P i ) for x i ∈ P i be an orbit series in P . Let Q = Q and Q i + = Orb ( x i , Q i ) . Using induction on i it is easy to see that P i ≤ Q i . Since Q ∈ t OS n , there exists a number k ≤ n such that Q k is the trivial quandle with one element. Since P k ≤ Q k , it is also a trivial quandlewith one element. Therefore P ∈ t OS n . (cid:3) Example 5.3.
The Takasaki quandle Alex ( Q , − ) is connected, therefore it belongs to the families OS n , OS ω , and OS . However, Alex ( Q , − ) has a subquandle Alex ( Z , − ) which does not belong to OS (see Example 4.9). This example shows that the classes OS n , OS ω and OS are not closedunder taking subquandles. Example 5.4.
Let Q n = Alex ( C n , − ) be the dihedral quandle of order 2 n so that Q n ∈ t OS n + (see Example 4.1). Therefore Q n belongs to the families t OS ω , t OS , OS ω and OS . However thequandle Q = ∏ n ∈ N Q n does not belong to OS (and, therefore, does not belong to t OS ω , t OS , OS ω or OS ). This example shows that the families t OS ω , t OS , OS ω , and OS are not closed undertaking arbitrary direct products. Lemma 5.5.
Let Q be a quandle and α ∈ Con ( Q ) . If Q / α ∈ t OS n (resp. Q / α ∈ OS n ) and [ a ] α ∈ t OS m (resp. [ a ] α ∈ OS m ) for all a ∈ Q , then Q ∈ t OS n + m (resp. Q ∈ OS n + m ).Proof. We provide the details for the case of Q / α ∈ t OS n and [ a ] α ∈ t OS m for all a ∈ Q ; the othercase is proved analogously. In this case the n -th element of every orbit series of Q is contained inan equivalence class of α , i. e., Q n ⊆ [ a ] α for some a ∈ Q . The subquandle [ a ] α belongs to t OS m and therefore Q m + n = ( Q n ) m is a one element set, so Q ∈ t OS m + n . (cid:3) Corollary 5.6.
The classes t OS ω , t OS , OS ω and OS are closed under taking extensions. he classes t OS ω , t OS , OS ω and OS are closed under taking extensions and they all containthe trivial quandles. Then Q belongs to one of these classes if and only if Q / λ Q does, since theclasses of λ Q are trivial quandles. Hence we can investigate such properties just for conjugationquandles: Corollary 5.7.
A quandle Q belongs to one of the classes t OS ω , t OS , OS ω or OS if and only if Q / λ Q (which is Conj − ( Inn ( Q )) , see section 2.2) belongs to t OS ω , t OS , OS ω or OS , respectively. Orbit series conditions and connected subquandles.
Let the class of quandles that donot have non-trivial connected subquandles be denoted by n C S . This class of quandles is closelyrelated to the families defined by the orbit series conditions. Proposition 5.8.
The class n C S is closed under taking subquandles, extensions and direct products.Proof. Clearly every subquandle of a quandle with no proper connected subquandle has no properconnected subquandles. Let { Q i ∶ i ∈ I } be a family of quandles with no connected subquandles,and Q = ∏ i ∈ I Q i . If M is a connected subquandle of Q , then its projection onto Q i is trivial forevery i ∈ I , and therefore ∣ M ∣ = M be a connected subquandle of a quandle Q . If Q / α has no non-trivial connected sub-quandles, then M ⊆ [ a ] α for some a ∈ Q . If [ a ] α has no non-trivial connected subquandles, then ∣ M ∣ = (cid:3) The following statement describes the family of quandles t OS in terms of its connected subquan-dles. Proposition 5.9. t OS = OS ∩ n C S .Proof. ( ⊆ ) Suppose that P is a nontrivial connected subquandle of Q and let x ∈ P . Then P iscontained in the intersection of the principal orbit series of x . Therefore, this orbit series can notstabilize on the one-element trivial quandle. ( ⊇ ) Assume that Q does not belong to t OS , i. e., there exists an orbit series Q ≥ Q ≥ . . . thatdoes not stabilize on the trivial quandle with one element. Since Q ∈ OS , this orbit series muststabilize, i. e., there exists a number k such that Q k = Q k + . But by definition, Q k + is an orbit in Q k , and therefore Q k is a nontrivial connected subquandle of Q . (cid:3) Since every finite quandle belongs to OS , Proposition 5.9 has the following corollary. Corollary 5.10.
Let Q be a finite quandle. Then Q ∈ t OS ω if and only if Q ∈ n C S . Note that the statement of Corollary 5.10 does not necessarily hold for infinite quandles. Forexample, the infinite dihedral quandle Alex ( Z , − ) does not satisfy the descending orbit series con-dition. It is a union of two orbits each of which is isomorphic to Alex ( Z , − ) , which allows us toconstruct an infinitely descending orbit series. However, this quandle has no non-trivial connectedsubquandles. Remark 5.11.
Each element ( a, b ) ∈ Q = Alex ( Z , − ) × Alex ( Q , − ) is contained in the connectedsubquandle { a } × Alex ( Q , − ) . But Alex ( Z , − ) is a homomorphic image of Q that is not in OS ,and therefore Q does not belong to OS . So, not even being a union of connected subquandles is asufficient condition for a quandle to be in OS .5.3. Descending series conditions, reductive and locally reductive quandles.
In this sec-tion we study connections between the families t OS and t OS n and the families of reductive, locallyreductive and solvable quandles. Proposition 5.12. R n ⊆ t OS n ⊆ LR n for every n ∈ N . roof. It is easy to check by induction that the n -th element of every orbit series of a quandle Q is contained in a single equivalence class of O nQ . According to Proposition 3.2, if Q is n -reductive,then O nQ = Q and therefore the n -th element of every orbit series of Q is trivial. Hence Q ∈ t OS n .If Q ∈ t OS , then it is 1-locally reductive. Assume that Q ∈ t OS n . Then the orbits belongto t OS n − and so by induction they are ( n − ) -locally reductive. By Lemma 3.5, Q is n -locallyreductive. (cid:3) Corollary 5.13.
Let G be a nilpotent group of nilpotency class n . Then Conj ( G ) ∈ t OS n .Proof. In this case Inn ( Conj ( G )) = G / Z ( G ) has nilpotency class n −
1, and therefore by Proposi-tion 3.2, Conj(G) belongs to R n . By Proposition 5.12 it belongs to t OS n . (cid:3) The following example shows that the inclusions in Proposition 5.12 are strict (except for thecase n = Example 5.14.
Let Q be the quandle with the following operation table: Q = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ The orbits of Q are (evidently) { , . . . , } , { , . . . , } and { , . . . , } . The first orbit is the directsum of two copies of Alex ( C , − ) , which is not trivial, and so Q ∈ t OS , but Q ∉ t OS . On theother hand, Q is 2-locally reductive, 5-reductive, and, critically, not 3-reductive. This example wasconstructed so that the action of the other two trivial orbits on Alex ( C , − ) + Alex ( C , − ) does notdisturb its 2-local reductivity.The next two examples establish the strictness of other inclusions. The first shows that R ω ≠ t OS ω and the second illustrates that LR ω / ⊆ t OS . Example 5.15.
Let G be a 3-Engel group that is not nilpotent (which exists by [36, Section 18.3])and let Q = Conj ( G ) . Since G is 3-Engel, it follows that ⟨ x G ⟩ is nilpotent of length 2 for every x ∈ G [35]. By Proposition 3.2, this means that the orbits of Q belong to R ⊆ t OS and by Corollary 3.5,it follows that Q ∈ t OS . On the other hand, G is not nilpotent, and therefore by Proposition 3.2 Q ∉ R ω . Example 5.16.
Let B = B ( , k ) be the free Burnside group of rank 2 and exponent 2 k with k largeenough (so that B is infinite). By the solution of the restricted Burnside problem, the group B hasonly finitely many subgroups of finite index { B i ∶ ≤ i ≤ m } and so B = ⋂ mi = B i is a finite-indexsubgroup of B [46]. The subgroup B has no finite-index subgroup and it has exponent dividing 2 k . oreover B is finitely generated, since it is a finite-index subgroup of a finitely generated group,and therefore B has an infinite simple quotient G , which also has exponent dividing 2 k .Let g ∈ G be an element of order two and ̂ g be the inner automorphism induced by g . Let Q = Alex ( G, ̂ g ) be the Alexander quandle of the group G with respect to the automorphism ̂ g .The operation in Q is given by x ⊳ y = ̂ g ( yx − ) x . Directly from the definition of the operation,Orb ( , Q ) = ⟨{̂ g ( x ) − x ∶ x ∈ G }⟩ . Since the subgroup ⟨{̂ g ( x ) − x ∶ x ∈ G }⟩ is normal in G (see, forexample, [3, Proof of Proposition 1] or [29]), and G is simple, it follows that ⟨{̂ g ( x ) − x ∶ x ∈ G }⟩ = G .Hence, Orb ( , Q ) = G = Q and so Q is connected. In particular, Q does not belong to t OS (nor t OS ω ).Now we’ll show that Q ∈ LR k + . For x ∈ Q let t ( x ) = ̂ g ( x ) − x = x ⊳
1. Since g has order two, foran arbitrary x ∈ G we have the equalities ̂ g ( t ( x )) = ̂ g (̂ g ( x ) − x ) = ̂ g ( x ) − ̂ g ( x ) = x − ̂ g ( x ) = t ( x ) − . We show that t n ( x ) = t ( x ) n − by induction on n . The basis of induction is trivial. Using the aboveequation for ̂ g ( t ( x )) and the induction hypothesis we have t n ( x ) = ̂ g ( t n − ( x )) − t n − ( x ) = ̂ g ( t ( x ) n − ) − t ( x ) n − = t ( x ) n − t ( x ) n − = t ( x ) n − which completes the induction. Translating this to the operation in Q , and considering the partic-ular case n = k +
1, we have: ( . . . (( a ⊳ ) ⊳ ) ⊳ . . . ) ⊳ ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k + = t k + ( a ) = t ( a ) k = a ∈ Q . Since Q is connected, apply the inner automorphism taking 1 to b to see that ( . . . (( a ⊳ b ) ⊳ b ) ⊳ . . . ) ⊳ b ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ k + = b holds for all a, b ∈ Q , i. e., Q is k + Theorem 5.17.
The following system of inclusions hold R n ⊂ t OS n ⊂ LR n ∩ ∩ ∩ R ω ⊂ t OS ω ⊂ LR ω and all inclusions in this system are strict. By Proposition 3.10, finite quandles are reductive if and only if they are locally reductive. Hence,Theorem 5.17 has the following corollary.
Corollary 5.18.
Let Q be a finite quandle. Then Q ∈ R ω ⇔ Q ∈ t OS ⇔ Q ∈ LR ω . If Q is medial then it is n -locally reductive if and only if it is n -reductive [31]. Hence, Theo-rem 5.17 has the following corollary. Corollary 5.19.
Let Q be a medial quandle, and n ∈ N . Then Q ∈ R n ⇔ Q ∈ t OS n ⇔ Q ∈ LR n . Abelian quandles are medial, so Corollary 5.19 can be applied to abelian quandles. The followingstatement partially extends Corollary 5.19 from abelian quandles to solvable quandles.
Proposition 5.20.
Let Q be a solvable quandle. Then Q ∈ t OS ω if and only if Q ∈ LR ω . Inparticular, if Q is k -locally reductive and its solvability length is n then Q ∈ t OS nk . roof. The claim is true for abelian quandles. Assume by induction that the claim is true forquandles of solvable length n . Let Q be k -locally reductive and solvable of length n + α = γ n ( Q ) be the n -th element of the derived series of Q . The factor Q ′ = Q / α is k -locally reductiveand n -solvable, so Q ′ ∈ t OS nk . All of the subquandles [ a ] α are abelian and k -locally reductive,therefore is in t OS k and so Q ∈ t OS nk + k ⊂ t OS ω by Lemma 5.5. (cid:3) Propositions 3.6 and 5.20 imply the following statement.
Corollary 5.21.
Let G be a solvable group. Then Conj ( G ) ∈ t OS ω if and only if G is an Engelgroup. Finally, we characterize the conjugation quandles in the class t OS . Proposition 5.22.
Let G be a group and H ⊆ G be a subset of G closed under conjugation. Then Conj ( H ) ∈ t OS if and only if the elements of H are -Engel elements of ⟨ H ⟩ .Proof. If x ∈ H , then the orbit of x in H is the set { h − xh ∶ h ∈ ⟨ H ⟩} . So, we have ( h − xh ) − x ( h − xh ) = h − x − hxh − xh = [ h, x ] x. Thus Conj ( H ) ∈ t OS if and only if [ h, x ] = h ∈ ⟨ H ⟩ . (cid:3) Proposition 5.22 has the following corollary.
Corollary 5.23.
Let G be a group and Q = Conj ( G ) . Then Q ∈ t OS if an only if G is a -Engelgroup. Moreover, in this case Q ∈ R .Proof. The fact that Q ∈ t OS if an only if G is a 2-Engel group follows from Proposition 5.22.Every 2-Engel group is nilpotent of length at most three [37]. Therefore Inn ( Q ) = G / Z ( G ) isnilpotent of length at most 2 and so according to Proposition 3.2 we have Q ∈ R . (cid:3) Note that this corollary gives us a bound on the length of reductivity for a quandle whose orbitsof orbits are trivial, at least in the case of conjugation quandles. This observation naturally leadsto the following unresolved questions, with which we conclude the paper.In general by Corollary 5.18, if a finite quandle Q is k -locally reductive, then it is in t OS j forsome j . Also, if it is in t OS j , then it is i -reductive for some i . Question 5.24.
Is it possible to give a bound on j in terms of k ? On i in terms of j ?Looking back at example 5.14, we see that when k =
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IMAS–CONICET and Universidad de Buenos Aires
E-mail address : [email protected] (A. Crans) Loyola Marymount University, Los Angeles, USA
E-mail address : [email protected] (T. Nasybullov) Sobolev Institute of Mathematics, Novosibirsk, Russia
E-mail address : [email protected] (G. Whitney) Studio Infinity, Los Angeles, USA
E-mail address : [email protected]@post.harvard.edu