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APPEARED IN BULLETIN OF THE

arXiv:math/9210221v1 [math.GR] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 257-260ON THE BURNSIDE PROBLEM ON PERIODIC GROUPSSergei V. IvanovAbstract. It is proved that the free m-generated Burnside groups B(m, n) of ex-ponent n are infinite provided that m > 1, n ≥248.In 1902 William Burnside posed the following problem [2].

Does a group G haveto be finite provided that G has a finite set of generators and its elements satisfythe identity xn = 1? In other words, must a finitely generated group G of exponentn be finite?In the same paper, Burnside proved that the problem was solved in the affir-mative for groups of exponents 2, 3 and for 2-generated groups of exponent 4 aswell.In 1940 Sanov [12] obtained a positive solution to the Burnside problem for thecase of exponent 4.The next significant step was made by Marshall Hall [4] in 1957 when he solvedthe problem in the affirmative for the exponent of 6.In 1964 Golod [3] found the first example of an infinite periodic group with a finitenumber of generators.

Although that example did not satisfy the identity xn = 1,i.e., the group was of unbounded exponent, it gave the first positive evidence thatthe Burnside problem might not be solved affirmatively for all exponents (and itmight possibly fail for very large exponents).In 1968 Novikov and Adian achieved a real breakthrough in a series of funda-mental papers [9] in which some ideas put forward by Novikov [8] in 1959 were de-veloped to prove that there are infinite periodic groups of odd exponents n ≥4381with m > 1 generators. Later, Adian [1] improved the estimate up to n ≥665(n is odd again).

Notice in the papers [9] that, in fact, the free Burnside groupsB(m, n) = Fm/Fnm, where Fm is a free group of rank m > 1 and Fnm is the normalsubgroup of Fm generated by all nth powers (with odd n ≥4381) of elements ofFm, were constructed and studied. Using a very complicated inductive construc-tion, Novikov and Adian presented the group B(m, n) by defining relations of the1991 Mathematics Subject Classification.

Primary 20F05, 20F06, 20F32, 20F50.Received by the editors January 7, 1992Lectures on results of this note were given at the University of Utah (October 30, 1991, January9, 16, 23, 30, 1992), the University of Wisconsin-Parkside (February 6, 1992), the City Universityof New York (February 7, 1992), the University of Nebraska-Lincoln (February 13, 14, 1992), theKent State University (March 2, 1992), the University of Florida-Gainesville (March 23, 1992).It is the pleasure of the author to thank these universities for sponsoring his visits as well as toexpress his gratitude to Professors G. Baumslag, B. Chandler, P. Enflo, S. Gagola, S. Gersten, J.Keesling, A. Lichtman, S. Margolis, J. Meakin, G. Robinson, and J. Thompson for their interestin this workc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2S. V. IVANOVform An = 1, where A’s are some specially chosen elements of Fm, and studiedtheir consequences.

They not only obtained the result that the group B(m, n) isinfinite but also other important information about B(m, n). For example, it wasproved that the word and conjugacy problems are solvable in B(m, n) and that anyfinite or abelian subgroup of B(m, n) is cyclic (under the restrictions on m and nabove; for these and other results see [1]).At the same time, it should be pointed out that [9] is very long and of verycomplicated logical structure.In 1982 Ol′shanski˘ı [10] succeeded in finding a considerably shorter proof of thetheorem of Novikov and Adian, although the estimate n > 1010 (where n again isodd) of [10] is much worse than n ≥665 of Adian’s [1].

On the other hand, it isworth noting that the approach of Ol′shanski˘ı’s to treat the free Burnside groupsB(m, n) is based on a powerful geometric method of graded diagrams (see [11, 6]for numerous applications of the method in combinatorial group theory).Thus, it is known that the Burnside problem is settled in the affirmative forexponents n = 2, 3, 4, 6 and in the negative for the exponents that have an odddivisor not less than 665 (the latter is an easy corollary of the theorem of Novikovand Adian). In particular, the Burnside problem still remains open for exponentsof the form n = 2k.

Besides, there is no approach to study the free Burnside groupsB(m, n) of even exponent n, even if n has a rather great odd divisor, and the onlyknown characteristic of these groups is their infiniteness (unlike the case of oddexponents n ≥665).Now let us mention an unpublished work [5] of the author’s where the freeBurnside groups B(m, 2n) and B(m, 4n) with odd n ≫1 were constructed bymeans of defining relations in order to prove solvability of the word and conjugacyproblems for these groups and to obtain a description of their finite subgroups. Allefforts to extend the techniques of [5] to study the groups B(m, 8n) with odd n ≫1,however, were unsuccessful.In the meantime, quite new techniques have been developed in order to constructand study the free Burnside groups B(m, n) with any n ≫1 regardless of theoddness of n. The key point of the techniques is in obtaining a complete descriptionof finite subgroups of the free Burnside groups B(m, n).

Therefore, in Theorem A,which gives the negative solution to the problem of Burnside’s for all rather greatexponents, we include this description.Theorem A. Let B(m, n) be the free Burnside group of rank m and exponent n,m > 1 and n ≥248.

Then(a) The group B(m, n) is infinite. (b) The word and conjugacy problems are solvable in B(m, n).

(c) Suppose n = 2kn0, where n0 is odd. If k = 0 (i.e.

n is odd) then any finitesubgroup of B(m, n) is cyclic. If k > 0 (i.e.

n is even) then any finite subgroup ofB(m, n) is isomorphic to a subgroup of a direct product of two groups, one of whichis a dihedral group of order 2n and the other is a direct product of several copies ofa dihedral group of order 2k+1. In particular, if n = 2k then any finite subgroup ofB(m, n) is just a subgroup of a direct product of several copies of a dihedral groupof order 2n.

(d) The center of the group B(m, n) is trivial.Now let us give an inductive construction of the group B(m, n) of any exponentn ≫1 by means by defining relations. Notice that this construction repeats (it is

ON THE BURNSIDE PROBLEM ON PERIODIC GROUPS3a surprise in itself!) a construction invented by Ol′shanski˘ı [10] for the case wheren is odd.On the set of all nonempty reduced words over an alphabet A = {a±11 , .

. .

, a±1m }(we assume Fm to be the free group over the alphabet A), we introduce a totalorder a1 ≺a2 ≺· · · such that |X| ≤|Y | implies X ⪯Y , where |X| denotes thelength of the word X.Now, for each i ≥1, we define a word Ai called the period of rank i to be thesmallest (in terms of the order “≺” introduced above) of those words over A whoseorders in the group B(i −1), given by the presentation(∗)B(i −1) = ⟨a±11 , . .

. , a±1m | |An1 = 1, .

. .

, Ani−1 = 1⟩,are infinite.Notice that it is not clear a priori whether Ai exists for each i or not. Noticealso that infiniteness of the free Burnside groups B(m, n) (under the restrictionson m, n above) follows from the next Theorem B, since a finite group cannot bepresented by infinitely many independent defining relations over a finite alphabet.Theorem B.

Suppose m > 1 and n ≥248. Then the period Ai of rank i doesexist for each i ≥1, i.e., the system {Ani = 1}∞i=1 is infinite.

Next, the system{Ani = 1}∞i=1 can be taken as an independent set of defining relations of the freeBurnside group B(m, n) and order of the period Ai of any rank i ≥1 is equal inB(m, n) to n exactly.The following theorem contains some basic technical results about finite sub-groups of the groups B(i−1) and B(m, n). Notice that one can derive the algebraicdescription of finite subgroups of B(m, n) given in Theorem A (proceeding by in-duction on the maximum of heights of words of a finite subgroup of B(m, n)) from(a)–(e) of the following.Theorem C. Let B(m, n) be the free Burnside group of rank m > 1 and exponentn ≥248, and suppose that F(Ai) is a maximal finite subgroup of the group B(i −1)given by (∗) with respect to the property that F(Ai) is normalized by the period Aiof rank i.

Next, denote by Ji a word such that the inclusions J2i , (JiAi)2 ∈F(Ai),hold in B(i −1) (if there exists no such word we simply put Ji = 1). Then thefollowing claims hold:(a) Any word W having finite order in B(i −1) is conjugate in B(i −1) toa word of the form Akj T for some integer k, j < i and T ∈F(Aj).Besides,conjugacy in B(i −1) of the words Ak1j1 T1 and Ak2j2 T2, where T1 ∈F(Aj1) andT2 ∈F(Aj2), j1, j2 < i, that are not equal in B(i −1) to the identity yields j1 = j2and k1 = ±k2 (mod n).

(Therefore, given a nontrivial word W, such a number j isdefined uniquely in B(m, n) as well as in B(i −1) and called the height of the wordW. )(b) F(Ai) is defined uniquely, embeds into B(m, n), consists of words whoseheights are less than i, and is a 2-group.

(c) Any finite subgroup of B(m, n) consisting of words of heights ≤i and con-taining a word of height i exactly is conjugate to a subgroup of the group generatedby Ai, Ji, and all words from F(Ai). (d) The subgroup F(Ai) of B(i −1) is normalized by Ji.

(e) The words Ji and Ai act on the subgroup F(Ai) of B(i −1) by conjugationsin the same way as some words V1 and V2 act respectively, where V1 and V2 are

4S. V. IVANOVsuch that the subgroup of B(i −1) generated by V1, V2 and by all words from F(Ai)is finite and the equation J2i = V 21 (as well as (JiAi)2 = (V1V2)2 provided Ji ̸= 1)holds in B(i −1).Let us conclude with some remarks about proofs of Theorems A, B, and C.First, in the case of odd n (this special case emerges as the simplest one wherethe finite subgroups F(Ai) are trivial for all i), proofs of Theorems A, B, and Cvirtually repeat the proofs of Ol′shanski˘ı’s [10].

In particular, we use a geometricinterpretation of deducibility of relations in a group from its defining relations (thisinterpretation is based on the notion of van Kampen diagrams, see [7]).On the other hand, the case where n is even requires much more delicate investi-gations of various properties of finite subgroups of the free Burnside group B(m, n).As a matter of fact, we point out that these properties of finite subgroups of groupsB(m, n) along with subgroups F(Ai) degenerate in the case of odd n, and so onecan say that the works [1, 9, 10] primarily deal with most general characteristics ofthe groups B(m, n).Finally, we mention that our estimate n ≥248 is rather rough and can be stronglyimproved at cost of complication of proofs.Added in proof . It has been known to the author that I. Lysionak, The infinity ofBurnside groups of exponents 2κ for κ ≥13 (preprint), announces an independentsolution of the Burnside problem for exponents of the form 2κ ≥213 based on theNovikov-Adian method.AcknowledgmentsThe author is grateful to Professors Steve Gersten and Alexander Ol′shanski˘ı forhelpful discussions and their encouragement.References1.

S. I. Adian, The Burnside problems and identities in groups, Moscow, Nauka, 1975.2. W. Burnside, On unsettled question in the theory of discontinuous groups, Quart.

J. PureAppl. Math.

33 (1902), 230–238.3. E. S. Golod, On nil-algebras and finitely residual groups, Izv.

Akad. Nauk SSSR.

Ser. Mat.28 (1964), 273–276.4.

M. Hall, Solution of the Burnside problem for exponent 6, Proc. Nat.

Acad. Sci.

U.S.A.43 (1957), 751–753.5. S. V. Ivanov, Free Burnside groups of some even exponents, 1987 (unpublished).6.

S. V. Ivanov and A. Yu. Ol′shanski˘ı, Some applications of graded diagrams in combinatorialgroup theory, London Math.

Soc. Lecture Note Ser., vol.

160 (1991), Cambridge Univ.Press, Cambridge and New York, 1991, 258–308.7. R. C. Lyndon and P. C. Schupp, Combinatorial group theory, Springer-Verlag, Heidelberg,1977.8.

P. S. Novikov, On periodic groups, Dokl. Akad.

Nauk SSSR Ser. Mat.

27 (1959), 749–752.9. P. S. Novikov and S. I. Adian, On infinite periodic groups I, II, III, Izv.

Akad. Nauk SSSR.Ser.

Mat. 32 (1968), 212–244; 251–524; 709–731.10.

A. Yu. Ol′shanski˘ı, On the Novikov-Adian theorem, Mat.

Sb. 118 (1982), 203–235.11., Geometry of defining relations in groups, Moscow, Nauka, 1989.12.

I. N. Sanov, Solution of the Burnside problem for exponent 4, Uchen. Zap.

Leningrad StateUniv. Ser.

Mat. 10 (1940), 166–170.Higher Algebra, Department of Mathematics, Moscow State University, Moscow119899 RussiaCurrent address: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112E-mail address: ivanov@math.utah.edu

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