Conjugacy closed loops and their multiplication groups

Transcription

1 Journal of Algebra 272 (2004) Conjugacy closed loops and their multiplication groups Aleš Drápal Department of Mathematics, Charles University, Sokolovská 83, Praha 8, Czech Republic Received 9 December 2002 Communicated by George Glauberman Abstract A loop Q is said to be conjugacy closed if the sets {L ; Q} and {R ; Q} are closed under conjugation. Let L and R be the left and right multiplication groups of Q, respectively, and let Inn Q be its inner mapping group. If Q is conjugacy closed, then there eist epimorphisms L Inn Q and R Inn Q that are determined by L R 1 L and R L 1 R. These epimorphisms are used to epose various structural properties of Mlt Q = LR Elsevier Inc. All rights reserved. Keywords: Conjugacy closed loop; Multiplication group; Nucleus The notion of the conjugacy closed loop seems to be one of the most natural generalizations of the group concept. Indeed, groups are those loops, in which the left translations L a (and also the right translations R a ) are closed under composition (L a L b is always equal to some L c ). Groups also satisfy a similar condition for conjugation (L a L b L 1 a is always equal to some L c ), and there eist loops which are not groups, and which possess left and right translations that are closed under conjugation. Loops with this property are called conjugacy closed. The concept of conjugacy closedness goes back to Soikis [12], and, independently, to Goodaire and Robinson [7]. Let Q be a conjugacy closed loop, and put L = L a ; a Q and R = R a ; a Q. Groups L and R generate the multiplication group Mlt Q, and its stabilizer InnQ = Work supported by institutional grant MSM and by Grant Agency of Czech Republic, Grant 201/02/ address: drapal@karlin.mff.cuni.cz /$ see front matter 2004 Elsevier Inc. All rights reserved. doi: /j.jalgebra

2 A. Drápal / Journal of Algebra 272 (2004) (Mlt Q) 1 is known as the inner mapping group. We shall show (Theorem 3.1) that there eists an epimorphism L Inn Q determined by L T = R 1L, and that its kernel is equal to Z(L) ={R a ; a Q} L. We shall also observe that L 1 = R 1 (Corollary 2.5) and that InnQ = T a ; a Q (Theorem 2.8). (The latter facts have easy direct proofs, but do not seem to have been eplicitly recorded before.) There has been an upsurge of interest in conjugacy closed loops recently, witnessed, e.g., by [10] and [9]. One of the motives causing this interest has been a seemingly open problem whether Q/N has to be an abelian group for every conjugacy closed loop. Here N means the nucleus; i.e., the subloop formed by all a Q that associate (in any of the three possible positions) with all,y Q. This problem was solved by Basarab [1] in a positive way, but his result does not seem to have been noticed within the western mathematics up to The first version of this paper (from May 2002 and before I knew Basarab s result) has already contained a proof that Q/N has to be a group (Theorem 3.5). But to prove its commutativity, I had to employ, following Basarab, the identity that epresses the universality of conjugacy closed loops (cf. Lemma 4.2). (A property of loops is said to be universal, if it is retained by all isotopic loops.) Nevertheless, the proof presented in this paper is still different from that of [1] it is not so direct, but it reveals more about the structure of the loop. Basarab s proof is restated in [9]. It applies to a broader class of loops, namely to universal left conjugacy closed loops (loops where left translations are closed under conjugation in every of its isotopes, cf. [11]). Some of the results in this paper remain true, in some form, in the situation of one-sided conjugacy as well (e.g., Theorem 3.1). My interest in conjugacy closed loops stems from the discovery that they appear naturally in the contet of the inverse problem for multiplication groups [4 6] (the problem which permutation groups can be obtained as a multiplication group of a loop). Sections 1 and 2 contain (up to few eceptions mentioned above) standard material. All the needed proofs are short, and hence the cost to make the paper self-contained has been low. The main purpose of this paper is to clarify relationships within multiplication groups of conjugacy closed loops, and the last statement of the paper (Proposition 4.6) describes the known facts from the viewpoint of abstract group theory. Eamples of conjugacy closed loops can be found in [7,8,10] and [9]. Mappings are composed in this paper from the right to the left, and hence T a means here Ra 1L a (and not L 1 a R a, which is more usual and which here is equal to Ta 1 ). 1. Loops and groups Loops can be described as binary systems possessing a unit element, where the equations a = b and ya = b have unique solutions. One usually sets = a\b and y = b/a, and then a loop becomes a system with three binary operations. The latter approach is useful when epressing loop properties by means of identities. Ecept for notation conventions, most results in this section can be found in the standard sources [2,3].

3 840 A. Drápal / Journal of Algebra 272 (2004) Let Q be a loop and let 1 be its unit. The set of all left translations L a : a generates a permutation group on Q that is known as the left multiplication group of Q. The right translations yield the right multiplication group, and we shall denote the multiplication group L a,r a ; a Q by Mlt Q. In group theory one shows how to obtain generators of the stabilizer G ω ={ϕ G; ϕ(ω) = ω}, whereg is a permutation group on Ω and ω Ω, given a generating set X and permutations g γ G with g γ (ω) = γ for each γ Ω. ThenG ω = g 1 (γ) g γ ; X and γ Ω, and this yields the well known description of generators of the inner mapping group InnQ = (Mlt Q) 1 : Proposition 1.1. Let Q be a loop, and let L and R be its left and right multiplication groups, respectively. Then: L 1 = L 1 y L L y ;,y Q, R 1 = R 1 y R R y ;,y Q, and InnQ is generated by L 1 R 1 {T ; Q}. Recall that T = R 1 L,forall Q. A congruence of a loop Q is completely determined by the block that contains the unit element, and this block has to be a subloop of Q. Subloops which give rise to a congruence are called normal, and it is not difficult to show that a subloop H is normal if and only if (yh) = (y)h, (H )y = H(y) and H = H,forall,y Q. The normal subloops can be thus characterized in this way: Proposition 1.2. A subloop H of a loop Q is normal if and only if InnQ acts on H. The concept of isotopy is a specific feature of loop theory. It often conveys unepected computational power, despite its geometrical origin and motivation. An isotopism of loops Q 1 and Q 2 is a triple (α,β,γ)of bijective mappings Q 1 Q 2, where α() β(y)= γ( y) for all,y Q 1. When Q 1 = Q 2, then one speaks about an autotopism. All autotopisms of a loop Q clearly form a group. The following lemma (which is well known) is of crucial importance. Lemma 1.3. Let Q be a loop and let α and β be permutations of Q such that (α,β,α) or (β,α,α) is an autotopism. If α(1) = 1,thenα = β and α is an automorphism. Proof. Suppose α(1) = 1. If (α,β,α) is an autotopism, then α()β(y) = α(y) for all,y Q. By setting = 1 one obtains β(y)= α(y),forally Q. Let Q be a loop. Sets N λ ={a Q; a(y) = (a)y for all,y Q}, N ρ ={a Q; (y)a = (ya) for all,y Q} and N µ ={a Q; (a)y = (ay) for all,y Q} are known as the left, right and middle nucleus, respectively. The set N = N λ N ρ N µ consists of all elements that associate with any other two elements, and is called the nucleus of Q.

4 A. Drápal / Journal of Algebra 272 (2004) Lemma 1.4. Let Q be a loop and a its element. Then: (i) a N λ (L a, id Q,L a ) is an autotopism, (ii) a N ρ (id Q,R a,r a ) is an autotopism, and (iii) a N µ (Ra 1,L a, id Q ) is an autotopism. Proof. The conditions for autotopisms can be epressed by (i) (a)y = L a () y = L a (y) = a(y), (ii) (ya)= R a (y) = R a (y) = (y)a,and (iii) (/a)(ay) = Ra 1()L a(y) = y, respectively. The latter condition turns out to be (ay)= (a)y when is replaced by a. Lemma 1.5. Let Q be a loop and let L and R be its left and right multiplication groups, respectively. Put G = Mlt Q. Then C G (R) ={L a ; a N λ } and C G (L) ={R a ; a N ρ }. Proof. Consider a Q and note that L a R (y) = a(y) is equal to R L a (y) = (ay) for all,y Q if and only if a N λ. Therefore L a C G (R) if and only if a N λ. If ϕ C G (R), thenϕr (1) = R ϕ(1) for all Q, and hence ϕ = L ϕ(1) is a left translation. Proposition 1.6. Let Q be a loop and let L and R be its left and right multiplication groups, respectively. Then Z(R) ={L a ; a N λ } R and Z(L) ={R a ; a N ρ } L. Furthermore, M λ ={a Q; L a Z(R)} Z(N λ ), M ρ ={a Q; R a Z(L)} Z(N ρ ), and a L a and a R a are isomorphisms M λ = Z(R) and Mρ = Z(L), respectively. Proof. The description of Z(R) follows from Lemma 1.5 immediately, and so M λ N λ. If a,b M λ,then(l a ) 1 = L a 1 Z(R), a 1 M λ,andl a L b = L ab Z(R), ab M λ. We see that M λ is a subgroup of N λ,andthata L a yields an isomorphism M λ = Z(R). Recall that a left translation in a group belongs to the right multiplication group of the group if and only if it is induced by a central element. By considering the left and right translations of N λ we hence obtain M λ Z(N λ ). Let Q be a loop and consider e,f Q. Define a new operation on Q by y = (/e) (f \y).thenfeis its unit, and so Q( ) is a loop. The triple (L 1 e,rf 1, id Q) yields an isotopism Q( ) Q( ), andq( ) is called a principal isotope of Q. Itiswellknown and easy to establish that every loop isotopic to Q is isomorphic to some of its principal isotopes. A loop Q is said to be a G-loop, if all its isotopes are isomorphic to Q. A principal isotope determined by (e, f ) is said to be a left principal isotope, iff = 1, and a right principal isotope, ife = 1. Let Q( ) be the left principal isotope of Q( )

5 842 A. Drápal / Journal of Algebra 272 (2004) determined by e, andletq( ) be the right principal isotope of Q( ) determined by f. Then y = (/e) ((f/e)\y), and we see that Q( ) is a principal isotope of Q( ). A loop Q is thus a G-loop if and only if all its left and right principal isotopes are isomorphic to Q. Now,α(y) = (α()/e) α(y) when (Re 1 α, α, α) is an autotopism, and β(y) = β() (f \β(y)) when (β, L 1 f β,β) is an autotopism. We can hence state (following Wilson [13]): Proposition 1.7. A loop Q is a G-loop if and only if (1) for every e Q there eists a mapping α such that (Re 1 α, α, α) is an autotopism; and (2) for every f Q there eists a mapping β such that (β, L 1 f β,β) is an autotopism. Let G be a transitive permutation group on Ω. A nonempty subset Γ Ω forms a block if for all g G one has either g(γ ) = Γ or g(γ ) Γ =.IfΓ is a block, then g(γ ) is also a block, for every g G. Suchblocksareconjugate to Γ and the set of all conjugate blocks, say B, forms a G-invariant partition of Ω. The group G acts on B and yields so a permutation group on B. A normal subloop H of a loop Q is a block of G = Mlt Q, and the action of G on Q/H yields Mlt(Q/H ). (More precisely, the action of G on Q/H gives a canonical homomorphism from G onto Mlt(Q/H ), and the kernel of this homomorphism is the subgroup of all elements of G which fi every coset in Q/H.) The following easy observation is the basis of the subsequent proposition, and that proposition will turn out to be of great importance for our approach to conjugacy closed loops. Lemma 1.8. Let G be a transitive permutation group on Ω, with a block Γ. If the subgroup of G that fies the block Γ is normal, then it fies every conjugate of Γ. Proposition 1.9. Let Q be a loop and L be its left multiplication group. Let H be a normal subloop of Q. If{ϕ L; ϕ(h) = H } is a normal subgroup of L, thenq/h is a group. Proof. For every loop there eist nontrivial members of the left multiplication group that fi the unit element, unless the loop is a group (cf. Proposition 1.1). Consider such permutations with respect to the loop Q/H. Each of them is induced by some ϕ L, ϕ(h) = H. However, every such ϕ fies each coset of Q/H, by Lemma 1.8, and hence it induces id Q/H. 2. Basic properties The first three statements in this section are (more or less) direct consequences of the definition of conjugacy closedness. Lemma 2.4 and a part of Theorem 2.8 (the observation that mappings T generate InnQ) seem to be new, and Corollaries 2.5 and 2.9 are their immediate consequences, respectively. Propositions 2.6 and 2.7 and Theorem 2.10 can be found in [7].

6 A. Drápal / Journal of Algebra 272 (2004) Lemma 2.1. Let Q be a loop and let,y,z be its elements. (i) If L L y L 1 = L z,thenz = (y)/; and (ii) if R R y R 1 = R z,thenz = \(y). Proof. Write L L y L 1 = L z as L L y = L z L and apply this equality to 1. Then y = z and the rest is clear. A loop Q is said to be conjugacy closed,ifl L y = L (y)/ L and R R y = R \(y) R for all,y Q. If Q is a conjugacy closed loop, then L 1 L yl = L \(y) and R 1R yr = R (y)/,for all,y Q. Say that a subset S of a group G is closed under conjugation,if 1 y S and y 1 S for all,y S. From Lemma 2.1 we see that a loop is conjugacy closed if and only if its left and right translations are closed under conjugation, respectively. Lemma 2.2. Let Q be a conjugacy closed loop with elements, y and z. Then: (i) L L y L 1 = L (y)/ and R R y R 1 = R \(y), (ii) Ry 1L R y = T and L 1 y R L y = T 1, L 1 = R y Rz 1 and R L y L 1 z R 1 (iii) L R y R 1 z (iv) L 1 R yr 1 z (v) L R y L 1 (vi) L 1 L = R \y R 1 \z = R y R 1 R yl = R \y R 1 \1 and R 1 and R L y R 1 and R 1 = L y L 1 z, L yl 1 z R = L y/ L 1 = L y L 1, L yr = L y/ L 1 1/. Proof. Point (i) reiterates the definition of conjugacy closed loops. To see the connection with (ii) note that L L z = L (z)/ L can be epressed as (zy) = ((z)/)(y),whichis thesameasl R y = R y T.FromT = Ry 1L R y = Rz 1L R z one gets L R y Rz 1 L 1 = R y Rz 1, as required by (iii). The rest of (ii) and (iii) is left right symmetric. Points (iv) (vi) follow from (iii) in a direct way. Corollary 2.3. Let Q be a loop, and denote by L and R its left and right multiplication groups, respectively. If Q is conjugacy closed, then L and R are normal subgroups of Mlt Q. Lemma 2.4. Let Q be a conjugacy closed loop with elements, y, z.then: (i) Ry 1R yr = (R 1 (ii) L 1 y L yl = (L 1 y L 1 y R 1 z/, R y)(r 1L R ) = L 1/(y) L 1 /(y) L 1 L y)(l 1 1/, and R L ) = R (y)\1 R 1 (y)\ R 1 \1. Proof. The points of the lemma are left right symmetric; we shall prove only point (i). It suffices to prove its first equality, since the second one follows from point (vi) of Lemma 2.2. However, the first equality can be reduced to R y = L 1 R yt, which holds by point (ii) of Lemma 2.2.

7 844 A. Drápal / Journal of Algebra 272 (2004) Corollary 2.5. Let Q be a conjugacy closed loop and let L and R be its left and right multiplication groups, respectively. Then L 1 = R 1. Proof. Combine Lemma 2.4 and Proposition 1.1. The identities of conjugacy closed loops belong to those identities that can be described by the eistence of certain autotopisms: Proposition 2.6. A loop Q is conjugacy closed if and only if both of the triples (T,L,L ) and (R,T 1,R ) are autotopisms, for every Q. Proof. Consider, for eample, the triples (T,L,L ). They yield an autotopism for all Q if and only if ( ) (y)/ (z) = (yz) for all,y,z Q. This is the same as L (y)/ L = L L y, and the rest is clear. Proposition 2.7. Let Q be a conjugacy closed loop and consider,y Q. Then L 1 y L L y = Ty 1T T y, Ry 1R R y = T y T 1 Ty 1, and both these mappings are automorphisms of Q. Proof. From Proposition 2.6 we see that the triple (Ty 1T T y,l 1 y L L y,l 1 y L L y ) is a composition of three autotopisms, and hence it is an autotopism itself. The mappings L 1 y L L y fi 1, by Proposition 1.1, and thus we can use Lemma 1.3. Theorem 2.8. Let Q be a conjugacyclosed loop and denoteby L and R its left and right multiplication groups, respectively. Then {T ; Q} generates InnQ and L 1 = R 1 Aut Q is a normal subgroup of InnQ. Proof. This is just a combination of Proposition 1.1, Corollaries 2.3 and 2.5, and Proposition 2.7. Corollary 2.9. A subloop H of a conjugacy closed loop Q is normal if and only if H = H for all Q. Theorem Let Q be a conjugacy closed loop. Then N λ = N ρ = N µ is a normal subloop of Q. This subloop contains a Q if and only if T a Aut Q. Proof. Consider Q and use the criteria of Lemma 1.4. Since (R 1,L, id Q ) is an autotopism if and only if (T 1,L 1,L 1 )(R 1,L, id Q ) = (L 1, id Q,L 1 ) is an autotopism, by Proposition 2.6, we see that N λ = N µ,andn µ = N ρ follows by symmetry. Now, a Q belongs to N λ if and only if the triple (Ra 1,T a,ra 1)(L a, id Q,L a ) = (T a,t a,t a ) is an autotopism, by Lemma 1.4 and Proposition 2.6. Thus a Q is in the nucleus if and only if T a Aut Q.

8 A. Drápal / Journal of Algebra 272 (2004) If L a centralizes the right multiplication group R,thenL L a L 1 = L (a)/ centralizes R Mlt Q as well (see Corollary 2.3). However, a N λ if and only if L a centralizes R, by Lemma 1.5, and so T (a) N λ for every a N λ. The left right symmetry yields T 1 (a) N ρ for every a N ρ, and the nucleus is thus really a normal subloop, by Corollary The homomorphism Theorem 3.1. Let Q beaconjugacyclosedloop. DenotebyL itsleftmultiplicationgroup. Then there eists a unique homomorphism Λ : L InnQ that maps L to T for each Q. This homomorphism is the identity on L 1, it is surjective and its kernel is equal to Z(L) ={R ; Q} L. Proof. We need to show that L ε L ε n n = id Q implies T ε T ε n n = id Q for all 1,..., n Q and ε 1,...,ε n { 1, 1}. The triples (T ε i i,l ε i i,l ε i i ) are autotopisms for every i,1 i n, by Proposition 2.6, and hence ( T ε T ε n n,l ε L ε n n,l ε L ε ) n n is an autotopism as well. Since the two right-hand members of this triple are assumed to equal id Q, it follows that T ε T ε n n = id Q as well, by Lemma 1.3. We have Λ(ϕ) = ϕ whenever ϕ = L 1 y L L y for some,y Q, by Proposition 2.7. Hence Λ(ϕ) = ϕ,forallϕ L 1, by Proposition 1.1. The homomorphism Λ is surjective, by Theorem 2.8. Every ψ L can be uniquely epressed as L a ϕ,whereϕ L 1 and a Q. Assume ψ Ker Λ. ThenT a = Ra 1L a = ϕ 1 L 1, which means R a L. We also have T a Aut Q, by Theorem 2.8, and hence a N, by Theorem Now, R a Z(L), by Proposition 1.6, and ψ = L a ϕ = L a L 1 a R a = R a. We have verified Ker Λ Z(L), andz(l) {R a ; a Q} L is clear from Lemma 1.5. On the other hand, if R a L, thenr a = L a Ta 1 L implies T a L 1, and so Λ(R a ) = Λ(L a )Λ(Ta 1 ) = T a Ta 1 = id Q. Lemma 3.2. Let Q be a conjugacy closed loop. Then T T y T 1 = T (y)/ for all,y Q. Proof. We have T T y T 1 = Λ(L L y L 1 ) = Λ(L (y)/) = T (y)/, by Theorem 3.1 and point (i) of Lemma 2.2. Proposition 3.3. Let Q be a conjugacy closed loop, and denote by N its nucleus, and by L its left multiplication group. Then Inn Q = (T L 1 ; Q) and InnQ Aut Q = (T L 1 ; N).Furthermore,InnQ Aut Q is a normal subgroup of InnQ. Proof. The mapping Λ is a surjective homomorphism L Inn Q, by Theorems 2.8 and 3.1. The required epression of InnQ therefore follows from L = (L L 1 ; Q) and from the fact that Λ is identical on L 1. The mappings of L 1 are automorphisms, by Theorem 2.8, and hence T L 1 Aut Q if and only if T Aut Q. However,this

9 846 A. Drápal / Journal of Algebra 272 (2004) happens eactly when N, by Theorem Now, for Q and a N one has T (T a L 1 )T 1 = T (a)/ L 1, by Lemma 3.2 and Theorem 2.8. Nothing more is needed, since (a)/ = T (a) belongs to N, by Theorem 2.10 and Proposition 1.2. Corollary 3.4. Let Q be a conjugacy closed loop with the nucleus N. ThenΛ 1 (InnQ Aut Q) = (L L 1 ; N) ={ϕ L; ϕ(n) = N}. Theorem 3.5. Let Q be a conjugacy closed loop and let N be its nucleus. Then Q/N is a group. Proof. We have Λ 1 (InnQ Aut Q) L, by Proposition 3.3. The result hence follows from Corollary 3.4 and Proposition 1.9. The net statement does not seem to bear any further consequences; at least in this paper. However, it is of some interest, since it offers an alternative proof of Corollary 2.5. Proposition 3.6. Let Q be a conjugacy closed loop and denote by N the least normal subgroup of InnQ that contains the set {T y T 1 y T ;,y Q}.ThenL 1 = N = R 1. Proof. The group L 1 is normal in Inn Q, by Theorem 2.8, and is generated by the mappings T 1 y T T y, by Theorem 3.1 and Proposition 1.1. Hence y T T y = Ty 1 ( Ty Ty 1 T ) Ty N and T y Ty 1 T ( = T y T 1 y T ) T y T 1 y L 1, T 1 and thus L 1 = N. The definition of N is left right symmetric, since T ( T 1 ( Ty T 1 y T T y T 1 y ) 1 = T 1 T y Ty 1, and ) T 1 = T y Ty 1 T 1 = Ry 1 R yr, by Proposition 2.7. Hence R 1 = N as well. Theorem 3.7. Let Q be a conjugacy closed loop with a nucleus N, and let L and R be the left and right multiplication groups of Q, respectively. For every a Q: L a R R a L T a L 1 = R 1. Denote by M the set of all such a Q.ThenM is a normal subloop of Q, M Z(N) and Q/M is a group. Proof. We immediately see that L a R Ra 1L a = T a R 1. This condition is left right symmetric, since L 1 = R 1, by Corollary 2.5. From Theorem 3.1 it follows a M R a Z(L). Hence M is a subloop of Z(N), by Proposition 1.6. If a M and Q, then T T a T 1 L 1,sinceL 1 InnQ. Now,T T a T 1 = T T (a), by Lemma 3.2, and so T (a) M. We see that M is normal, by Corollary 2.9.

10 A. Drápal / Journal of Algebra 272 (2004) Observe now that for every a Q and ϕ L 1 the permutation Λ(L a ϕ) = T a ϕ belongs to L 1 if and only if a M. This means that {ψ L; ψ(m) = M} equals Λ 1 (L 1 ),which is a normal subgroup of L. It follows that Q/M is a group, by Proposition 1.9. Corollary 3.8. Let L and R be the left and right multiplication groups of a conjugacy closed loop Q, respectively. Then L R = Z(L)L 1 = Z(R)R 1,whereZ(L) ={R a ; a Q} L and Z(R) ={L a ; a Q} R.Furthermore,R a L a yields an isomorphism Z(L) = Z(R). Proof. A permutation L a ϕ, a Q and ϕ L 1, belongs to L R if and only if L a R,by Corollary 2.5, and so L a ϕ L R if and only if a M. However,Z(R) ={L a ; a M}, by Theorem 3.1, and L a a is an isomorphism Z(R) = M, by Proposition 1.6. The associator subloop of a loop Q is the least normal subloop of Q, saya, such that Q/A is a group. Corollary 3.9. The center of nucleus contains the associator subloop, in every conjugacy closed loop. This is a direct consequence of Theorem 3.7. In fact, Corollary 3.9 holds in every loop with a normal nucleus, factor over which is a group, as follows from [9, Lemma 4.2]. Nevertheless, Theorem 3.7 is a stronger statement since the subloop M can be a proper subgroup of Z(N). Indeed, suppose that N is abelian and Q : N =2. Pick Q \ N, put d = 2 and denote by α the restriction of T to N. Thenα becomes a nontrivial involutory automorphism of N, α(d) d, L 1 = R 1 is generated by the mapping a a, a dα(d 1 )a, a N, and the inner mapping T b, b N, isgivenbya a, a bα(b 1 )a. Hence b M if and only if bα(b 1 ) is a power of dα(d 1 ).Sinceaconjugacy closed loop Q of the above properties eists for every pair (α, d),whereα Aut N, d N, α 2 = id N,andα(d) d, one can clearly find many cases when bα(b 1 ) is not a power of dα(d 1 ). 4. Isotopy and commutativity We have seen in Section 3 that Q/N is a group, for every conjugacy closed loop Q. Basarab [1] proved that this group is always abelian. He uses the identity of Lemma 4.2 (see below) to generate additional autotopisms and then applies the fact that for every autotopism (α, id Q,β) we have α = β and α(1) N λ. The approach here is somewhat different. We start from the same identity, but instead of using autotopisms we interpret it as a generalization of the identity Ry 1L R y = R 1L (cf. Lemmas 2.2 and 4.3). The generalized identities allow us to derive easily \((y)/y) N, and this yields an alternative proof of Basarab s theorem, since we already know that Q/N is a group (Theorem 3.5). We start by a well known fact. Proposition 4.1. Every conjugacy closed loop is a G-loop. Proof. From Proposition 2.6 it follows that Proposition 1.7 can be used with α = L e and β = R f.

11 848 A. Drápal / Journal of Algebra 272 (2004) Lemma 4.2. Let u,, y and z be elements of a conjugacy closed loop. Then (u\(yz) ) = (( (u\y) ) /(u\) ) (u\(z) ). Proof. Conjugacy closed loops satisfy the identity (yz) = (y/)(z). This identity holds for the operation y = (u\y) as well, by Proposition 4.1, and yields the identity of the lemma when is replaced by and z is replaced by uz. The identity of Lemma 4.2 can be also epressed as Now, R u\(z) R 1 u\ = L 1 u L L 1 u R z = R u\(z) Ru\ 1 L L 1 u. R zr 1L u, by point (iv) of Lemma 2.2, and so L u L L 1 u R z = R z R 1 L ul L 1 u. By inverting this identity and using the left right symmetry we obtain: Lemma 4.3. Let, y and z be elements of a conjugacy closed loop. Then Ry 1 L (z)/zr y = R 1 L (z)/z and L 1 y R z\(z)l y = L 1 R z\(z). Proposition 4.4. Let Q be a conjugacy closed loop and denote by N its nucleus. Then [ L,L 1 y ] = L\((y)/y) and [ R,Ry 1 ] = R(y\(y))/, for all,y Q. The elements \((y)/y) and (y\(y))/ belong to the nucleus N. Proof. Fi,z Q. Wehave L R y L 1 = L (z)/z R y L 1 (z)/z = R yr 1, for all y Q, and hence L 1 L (z)/z =[L,L 1 z ] commutes with all R y, y Q. This means [L,L 1 z ] C Mlt Q (R),andso [ L,L 1 ] z = Lu for some u N, by Lemma 1.5. We have u = L 1 L (z)/z(1) = \((z)/z), andtherest follows by left right symmetry. Corollary 4.5 [1]. Let Q be a conjugacy closed loop and let N be its nucleus. Then Q/N is an abelian group.

12 A. Drápal / Journal of Algebra 272 (2004) Proof. Since we know that Q/N is a group (by Theorem 3.5), it suffices to observe that the commutator [N,(yN) 1 ] equals N = 1 Q/N,forall,y Q. However, this follows from Proposition 4.4. Note that Theorem 3.5 is not essential for the proof of Corollary 4.5 since one sees immediately that a commutative conjugacy closed loop has to be an abelian group. Kinyon, Kunen and Phillips proved recently that L 1 is an abelian group [9, Corollary 4.10]. Their proof is quite short and relies on the fact that all associators belong to Z(N). We shall use their result in the following concluding survey of relationships within Mlt Q. Proposition 4.6. Let Q be a conjugacy closed loop and let L and R be its left and right multiplication groups, respectively. Put G = Mlt Q and H = InnQ.PutK = L 1 = L H. Then K = R 1 = R H and K is abelian. Furthermore: (i) L G, R G, G = LR, L/Z(L) = H = R/Z(R), Z(L) = Z(R), and L R = Z(L) K = Z(R) K; (ii) G/R = H/K = G/L = L/L R = R/L R and G/(L R) = (L/L R) (R/L R); (iii) N L (K) L, N R (K) R and the group N = N L (K)H = N R (K)H is normal in G; (iv) G/ N = L/N L (K) = R/N R (K) is an abelian group which is isomorphic to Q/N, N the nucleus of Q. Proof. For point (i) use Theorem 3.1 and Corollary 3.8. We have G/R = H R/R = H/H R = H/K = H/H L = H L/L = G/L = LR/L = L/L R, and this makes (ii) clear. Now, N L (K) consists of all L a ϕ,whereϕ K = L 1 and a Q is fied by every ψ L 1, which means a N ρ = N. The loop Q/N is a group, and hence both ϕ and L a move no coset of N. The group N L (K) therefore consists of all mappings α L that move no coset of N. Similarly, N consists of all α G that move no coset of N. Wehave G/ N = L N/ N = L/ N L and N L = N L (K)(H L) = N L (K)K = N L (K). The points (iii) and (iv) are thus clear, since G permutes the blocks of N in a regular way, by Corollary 4.5. Acknowledgment I thank to Michael Kinyon, who read the paper and suggested changes which improved its organization and clarity. References [1] A.S. Basarab, Klass LK-lup, Mat. Issled. 120 (1991) 3 7.

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