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On Nuclearly Nilpotent Loops of FiniteExponentPeter Plaumann a & Liudmila Sabinina ba Mathematics Institute , University Erlangen-Nürnberg , Erlangen,Germanyb Faculty of Science , Autonomous University of Morelos State ,Morelos, MexicoPublished online: 20 Jun 2008.

To cite this article: Peter Plaumann & Liudmila Sabinina (2008) On Nuclearly Nilpotent Loops of FiniteExponent, Communications in Algebra, 36:4, 1346-1353, DOI: 10.1080/00927870701864072

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Communications in Algebra®, 36: 1346–1353, 2008Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870701864072

ON NUCLEARLY NILPOTENT LOOPSOF FINITE EXPONENT

Peter Plaumann1 and Liudmila Sabinina21Mathematics Institute, University Erlangen-Nürnberg,Erlangen, Germany2Faculty of Science, Autonomous University of Morelos State,Morelos, Mexico

We show that the restricted Burnside problem has a positive answer for suitable classesof nuclearly nilpotent loops. Using this technique we give a positive answer to therestricted Burnside problem for Moufang A-loops.

Key Words: A-loops; Burnside problem; Moufang loops.

2000 Mathematics Subject Classification: 20N05.

For a loop L, that is not necessarily power-associative, we say that L hasexponent n if all words of length n with a single letter x ∈ L in arbitrary bracketingare trivial in L. We say that L has rank at most d, where d is a natural number, if inL there is a generating subset containing at most d elements. In this case we writerk�L� ≤ d. If L is not finitely generated, we put rk�L� = �. If � is a class of loops,we denote by �n

d the class of all elements of � having exponent n and rank at mostd. Furthermore, we denote by �d the class of all L ∈ � such that rk�L� ≤ d and weput �∗

d = ⋃�n=1 �

nd . Finally, we set �n

∗ = ⋃�d=1 �

nd .

The following is a convenient way to describe the Burnside problem for classesof loops (see Kostrikin, 1990, I.1). Call a class � of loops a B-class if for every pair�n� d� of natural numbers there are constants �n�d��� ∈ � such that �L� ≤ �n�d���

for all L ∈ �nd . If � is a B-class, this means that the Burnside problem has a positive

solution for �. It is easy to see that the class of all abelian groups is a B-class, but ittook considerable effort to show that the class of all groups is not a Burnside class.

In order to describe the restricted Burnside problem, we call a class � ofloops an RB-class if for every pair �n� d� of natural numbers there are constants�n�d���∈� such that �E� ≤ �n�d��� for all finite loops E ∈ �n

d . If � is a B-class or aRB-class of loops, we call

(�n�d���

)�n�d�∈�×�

, respectively,(�n�d���

)�n�d�∈�×�

Burnsideconstants for �.

Received October 6, 2006; Revised March 8, 2007. Communicated by I. P. Shestakov.Address correspondence to Peter Plaumann, Mathematisches Institut, Universität Erlangen-

Nürnberg, Bismarckstrasse 1 1/2, Erlangen D-91054, Germany; Fax: +52-777-3297040; E-mail: peter.plaumann@mi.uni-erlangen.de

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ON NUCLEARLY NILPOTENT LOOPS OF FINITE EXPONENT 1347

For the class � of all groups, the restricted Burnside problem was settled whenZelmanov showed that � is an RB-class (Zelmanov, 1990, 1991).

The first important result for the Burnside problem for proper Moufang loopsis due to Bruck who has shown that in the class � of commutative Moufang loopsevery finitely generated loop without elements of infinite order is finite (Bruck, 1958,Theorem 11.3, p. 158). Grishkov more generally has treated the restricted Burnsideproblem for Moufang loops of prime exponent. For a natural number n we denoteby ��n� the variety all Moufang loops of exponent n. Grishkov (1987) has shownthat for a prime p �= 3, the class ��p� is an RB-class. This result was completed byNagy (2001) where it is shown that ��3� is even a B-class.

It should be remarked that when leaving the class of Moufang loops, thesituation is completely different; in Nagy (2001) it is shown that the Bol loops ofexponent 2 do not form an RB-class.

In this note we present a reduction principle (Lemma 1) which allows us totreat the Burnside problems for certain classes of loops L which have a normalsubloop N such that N and L/N belong to a B-class, respectively, and an RB-classof loops. We apply this principle to the class � of Moufang A-loops and to theclass � of all conjugacy closed loops and show that in these classes, the restrictedBurnside problem has a positive answer. For fundamental facts from the theory ofloops, we refer the reader to Bruck (1958) and Pflugfelder (1990).

1. A REDUCTION PRINCIPLE

If � and � are two classes of loops, we denote by �� the class of all loopsL which contain a normal subloop N ∈ � such that L/N ∈ �. Let � be a functionthat assigns to every loop L a characteristic normal subloop ��L�. Then ����� isthe class of all loops L for which ��L� ∈ � and L/��L� ∈ �.

For our reduction lemma, we consider the following setup ��:

(a) ��� classes of loops;(b) A function � assigning to every loop L a characteristic normal subloop ��L�;(c) A subclass � of the class �����;(d) A function � �×� −→ �×� such that for all C ∈ �n

d the loop ��C� hasrank at most ��n� d�.

Lemma 1 (Reduction Lemma). Assume ��. Then:

(i) If � and � are B-classes, then � is a B-class with Burnside constants�n���n�d�

����n�d���;(ii) If � and � are RB-classes, then � is a RB-class with Burnside constants

�n���n�d�����n�d���.

Proof. In order to show (i), let �n�d = �n�d��� and �n�d = �n�d��� be Burnsideconstants for � and �, respectively. Consider a loop C ∈ � which has exponent nand rank at most d. Since � ⊆ �����, it follows from �� that

�C/��C�� ≤ �n�d�

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1348 PLAUMANN AND SABININA

rk���C�� ≤ ��n� d��

���C�� ≤ �n���n�d�

Hence �C� = ���C�� · �C/��C�� ≤ �n�d�n���n�d�.

The proof of (ii) is the same if one restricts the attention to finite loopsin �. �

2. THE ASSOCIATOR SUBLOOP AND THE NUCLEUS

In any loop L, the associator �x� y� z� of elements x� y� z ∈ L is defined by theequation

�xy�z = �x�yz���x� y� z�

The subloop �L� L� L� generated by all associators in L is called the associatorsubloop of L. Obviously, �L� L� L� is a characteristic subloop of L, i.e., everyautomorphism of L fixes �L� L� L�.

The set of all elements u ∈ L satisfying �u� x� y� = �x� u� y� = �x� y� u� = 1 forall x� y ∈ L form a characteristic subloop N�L� which is called the nucleus of L.

In general, neither the associator subloop nor the nucleus of a loop L is normalin L. A factor of a loop L is an epimorphic image of a subloop of L. We call a loopL regulated if for every factor H of L the loops �H�H�H� and N�H� are normal in H .

The following lemma is a corollary of Lemma 4.2 in Kinyon et al. (2004).We give a direct proof here.

Lemma 2. Let L be a loop in which the nucleus N is a normal subloop. Forx� y� z ∈ L and u ∈ N�L�, the following identities hold:

(a) uxy = �uy�x, where ux ∈ N�L� is defined by xu = uxx for u ∈ N�L� and x ∈ L;(b1) �ux� y� z� = �x� y� z�;(b2) �x� uy� z� = �x� y� z�;(b3) �x� y� uz� = �x� y� z�.

Proof. Since u lies in the nucleus of L, one has

uxy�xy� = �xy�u = x�yu� = x�uyy�

= �xuy�y = ��uy�xx�y = �uy�x�xy�

Thus (a) holds.Using

��ux�y�z = �u�xy��z = u��xy�z��

�ux��yz� = u�x�yz���

one obtains (b1).

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ON NUCLEARLY NILPOTENT LOOPS OF FINITE EXPONENT 1349

Since N is a normal subloop of L, one has

�x�uy��z = ��xu�y�z = ��uxx�y�z = ux��xy�z��

x��uy�z� = x�u�yz�� = �xu��yz� = �uxx��yz� = ux�x�yz��

It follows that (b2) holds.In the same way, using (a) one shows

�xy��uz� = uxy��xy�z���

x�y�uz�� = uxy�x�yz���

which implies (b.3). �

Theorem 3. Let L be a loop with a normal nucleus N�L�. If L/N�L� is finite, then�L� L� L� is generated by at most �L/N�L��3 elements.

Proof. Let T be a system of representatives for the cosets of N�L� in L. By Lemma 2the set ��x� y� z� � x� y� z ∈ T� coincides with the set of all associators of L. �

3. NUCLEAR NILPOTENCY

For loops, even for finite loops, there are different notions of nilpotency. In thespirit of Bruck (1958, VI.1, pp. 94–96) we introduce a concept of nuclear nilpotency.

Let � be a class of loops which is closed under taking subloops andepimorphic images. Let now N�L� denote the normal closure of the nucleus of L forall L ∈ �. For L ∈ � we define the ascending nuclear series

N0�L� ⊆ N1�L� ⊆ · · · ⊆ Ni�L� ⊆ Ni+1�L� ⊆ · · ·

of normal subloops by N0�L� = 1� Ni+1�L�/Ni�L� = N(L/Ni�L�

). We say that the

loop L ∈ � is nuclearly nilpotent or for short �-nilpotent of class c, if Nc�L� = L, butNc−1�L� �= L. If L is �-nilpotent, the number ��L� = c is called the nuclear length of L.

Nuclear nilpotency is a generalization of central nilpotency (Bruck, 1958, p. 94).

Proposition 4. Every centrally nilpotent loop L of class c is nuclearly nilpotentsatisfying ��L� ≤ c.

Proof. Consider the upper central series �Zi�ci=0 of L (Bruck, 1958, VI.1, pp. 94–97).

One has Z1 ⊆ N1�L�. Hence Zi ⊆ Ni�L� for all i ∈ � by induction. Since L iscentrally nilpotent of class c, it follows that L = Zc ⊆ Nc�L�. �

4. NUCLEARLY NILPOTENT LOOPS OF FINITE EXPONENT

In this section we consider classes of loops satisfying the followingconditions:

(N1) All loops L ∈ are regulated;(N2) There is c ∈ � such that every L ∈ is nuclearly nilpotent of length ��L� ≤ c.

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1350 PLAUMANN AND SABININA

If the class satisfies these conditions, we say that it is a class of type ��c�.

Theorem 5. Let � be a class of type ��c�. Then � is an RB-class.

Proof. Putting ��i� = �L ∈ � � ��L� ≤ i�, one has ��c� = � We proceed by induction over c. For c = 1 the class � coincides with the class

� of all groups which is an RB-class (Zelmanov, 1990, 1991). Let �n�d be Burnsideconstants for �.

Assuming that the theorem is true for the classes ��i�� 1 ≤ i ≤ c − 1 onehas Burnside constants �n�d�i� for these classes. Take a finite loop L ∈ �n

d. ThenL/N�L� ∈ (

��c − 1�)nd. Hence one has �L/N�L�� ≤ �n�d�c − 1�. Applying Lemma 3

and putting d1 = ��n�d�c − 1��3, we obtain the following statement:

For every pair d� n ∈ � there is d1 ∈ � such that rk�L� L� L� ≤ d1 for all L ∈ �nd

(1)

As L/Nc−1�L� is a group, one sees that �L� L� L� ⊆ Nc−1�L�. From Schreier’ssubgroup theorem (cf. Huppert, 1967, 19.10 Satz, p. 141), applied to the groupL/�L� L� L� and its subgroup Nc−1�L�/�L� L� L�, it follows that

For d� n ∈ � there is d2 ∈ � such that rk�Nc−1�L��/�L� L� L� ≤ d2 for all L ∈ �nd

(2)

From (1) and (2) we obtain

rk(Nc−1�L�

) ≤ rk�L� L� L�+ rk(Nc−1�L�/�L� L� L�

) = d1 + d2 for all L ∈ �nd (3)

Hence, putting � = ���c − 1���� = � and ��L� = Nc−1�L� for all L ∈ �, thehypothesis �� of the Reduction Lemma is satisfied. �

5. SPECIAL CLASSES OF LOOPS

For an arbitrary loop one denotes by

R�a� = �x → xa� L −→ L�

L�a� = �x → ax� L −→ L

the right, respectively, left multiplications of L. The group = L generated by themappings

R�a� b� = R�a�R�b�R�ab�−1� L�a� b� = L�a�L�b�L�ba�−1�

T�a� = R�a�L�a�−1

is called the inner mapping group of L; the elements of L are called inner mappingsof L. A subloop K of L is normal in L if and only if H = H holds (see Bruck,1958, p. 61).

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ON NUCLEARLY NILPOTENT LOOPS OF FINITE EXPONENT 1351

Bruck and Paige (1956) have called a loop L an A-loop if every inner mappingof A is an automorphism. More specifically, a loop L is called an Al-loop if allmappings L�a� b� are automorphisms of L, and it is called an At-loop if all mappingsT�a� are automorphisms (see Carillo–Catalán and Sabinina, 2006).

A loop L is conjugacy closed if the sets �R�a� � a ∈ L� and �L�a� � a ∈ L� areclosed under conjugation within the group L (see Drapal, 2004).

We now collect information on the normality of the associator and the nucleusin some classes of loops.

Proposition 6. Let L be a loop.

(1) If L is an A-loop, then N�L� and �L� L� L� are normal subloops of L.(2) Every conjugacy closed loop is regulated.(3) If L is a Moufang loop, then N�L� is normal in L.(4) N�L� is normal in L if and only if T�x�N�L� = N�L� for all x ∈ L. In particular, if

L is an At-loop, then N�L� is a normal subloop of L.

Proof. (1) In an A-loop, every characteristic subloop is normal (Bruck and Paige,1956, 2.2 Theorem (ii), p. 309).

(2) The statement follows from Drapal (2004, Theorem 2.10) and Kinyonand Kunen (2006) (Lemma 2.5).

(3) This is just Bruck (1958, Theorem 2.1, p. 114).

(4) In every loop L, one has

x�yN�L�� = �xy�N�L�� �N�L�x�y = N�L��xy��

which implies

L�x� y�N�L� = N�L� = R�x� y�N�L�

for all x� y ∈ L. If L is an At-loop, the equality

L�x�N�L� = xN�L� = N�L�x = R�x�N�L�

holds for all x ∈ L, implying T�x�N�L� = N�L�. �

Theorem 7. The class � of all conjugacy closed loops is an RB-class.

Proof. By Drapal (2004, Theorem 3.5) every loop in � is nuclearly nilpotentof class ≤2. Hence by Proposition 6(2) the class � is a class of type ��2�.The proposition follows from Theorem 5. �

From Bruck and Paige (1956, p. 309) one obtains the following classificationsof Moufang Al-loops and Moufang At-loops.

Proposition 8. A Moufang loop L is an Al-loop if and only if L/N�L� is commutative.

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1352 PLAUMANN AND SABININA

Proposition 9. A Moufang loop L is an At-loop if and only if(L/N�L�

)3 = 1.

Using the notation � for the class of all Moufang A-loops and rememberingthat for any class � of loops we denote by �d the class of all L ∈ � with rk�L� ≤ dwe obtain the following theorem.

Theorem 10. The classes �d are of type ��d�.

Proof. Consider L ∈ �d. Then N�L� and �L� L� L� are normal in L (cf. Proposition6). By Proposition 8 and the theorem of Bruck and Slaby (Bruck, 1958, Theorem10.1, p. 157; Smith, 1978) the loop L/N�L� is centrally nilpotent of class ≤d − 1. Itfollows from Proposition 4 that ��L� ≤ d. �

The following result is a consequence of Theorems 5 and 10.

Corollary 11. The classes of Moufang A-loops is an RB-class.

ACKNOWLEDGMENTS

The authors thank E. Zelmanov for leading their attention to the Burnsideproblem for Moufang loops and for helpful discussions and to A. Drapal for usefulinformations.

Furthermore, the first author thanks PROMEP for financial support and thesecond one UCMEXUS-CONACyT, Grant 050011, CONACyT, Grant C2-44100,the University of California, and the Universität Erlangen-Nürnberg. Our specialthanks go to Tatiana Shvedova.

REFERENCES

Bruck, R. B. (1958). A Survey of Binary Systems. (Ergebnisse der Mathematik und ihrerGrenzgebiete. Neue Folge. Heft 20.) Berlin-Heidelberg: Springer-Verlag.

Bruck, R. B., Paige, L. J. (1956). Loops whose inner mappings are automorphisms. Ann.Math. 63:308–323.

Carillo–Catalán, R., Sabinina, L. (2006). The theory of Kikkawa spaces. In: Proceedings inNon-Associative Algebra and Their Applications. Baton Rouge, LA: Chapman & Hall.

Drapal, A. (2004). Conjugacy closed loops and their multiplication groups. J. Algebra272:838–850.

Grishkov, A. N. (1987). The weak Burnside problem for Moufang loops of prime period.Sib. Math. J. 28(3):401–405.

Huppert, B. (1967). Endliche Gruppen I. Berlin-New York: Springer-Verlag.Kinyon, M. K., Kunen, K. (2006). Power-associative, conjugacy closed loops. J. Algebra

304:679–711.Kinyon, M. K., Kunen, K., Phillips, J. D. (2004). Diassociativity in conjugacy closed loops.

Commun. Algebra 32:767–786.Kostrikin, A. I. (1990). Around Burnside. Transl. from the Russ. by James Wiegold.

(Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 20.) Berlin: Springer-Verlag.

Nagy, G. P. (2001). Burnside problems for Moufang and Bol loops of small exponent. ActaSci. Math. 67(3–4):687–696.

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ON NUCLEARLY NILPOTENT LOOPS OF FINITE EXPONENT 1353

Pflugfelder, H. O. (1990). Quasigroups and Loops: Introduction. Sigma Series in PureMathematics, 7. Berlin: Heldermann Verlag.

Smith, J. D. H. (1978). A second grammar of associators. Math. Proc. Camb. Philos. Soc.84:405–415.

Zelmanov, E. I. (1990). Solution of the restricted Burnside problem for groups of oddexponent. (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 54(1):42–59.

Zelmanov, E. I. (1991). Solution of the restricted Burnside problem for 2-groups. (Russian).Mat. Sb. 182(4):568–592.

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