1 Lecture ab out E m Zelmanov E m Zelmanov erhielt die Fields-Medaille fur seine brilliante Losung des lange Zeit o enen Burnside-Problems. Dab ei handelt es sich um ein tie iegen- des Problem der Grupp entheorie, der Basis fur das mathematische Studium von Symmetrien. Gefragt ist nach einer Schrankefur die Anzahl der Sym- metrien eines Ob jektes, wenn jede einzelne Symmetriebeschrankte Ordnung hat. Wie alle b edeutenden mathematischen Resultate hab en auch jene von Zelmanov zahlreiche Konsequenzen. Darunter sind Antworten auf Fragen, b ei denen ein Zusammenhang mit dem Burnside-Problem bis dahin nicht einmal vermutet worden war. Vor der Losung des Burnside-Problems hatte Zelma- nov b ereits wichtige Beitrage zur Theorie der Lie-Algebren und zu jener der Jordan-Algebren geliefert; diese Theorien hab en ihre Ursprunge in Geometrie bzw. in Quantenmechanik. Einige seiner dort erzielten Ergebnisse waren fur seine grupp entheoritischen Arb eiten von ausschlaggeb ender Bedeutung. Auf diese Weise wird einmal mehr die Einheit der Mathematik dokumentiert, und es zeigt sich, wie sehr scheinbar weit auseinanderliegende Teilgebiete mitein- ander verbunden sind und einander b eein ussen. Auszug aus der Laudatio von Walter Feit. E m Zelmanov has received a Fields Medal for the solution of the restric- ted Burnside problem. This problem in group theory is to a large extenta problem in Lie algebras. In proving the necessary prop erties of Lie algebras, Zelmanov built on the work of many others, though he went far b eyond what had previously b een done. For instance, he greatly simpli ed Kostrikin's results which settled the case of prime exp onent and then extended these metho ds to handle the prime power case. k The results from Lie algebras whichwork for exp onent p with p an o dd k prime are not adequate for the case of the exp onent2 . Zelmanovwas the rst to realize that in this case the theory of Jordan algebras is of great signi cance. 1 This do cument has b een repro duced from DMV-Mitteilungen 4/94 with friendly p er- mission from Birkhaeuser Verlag 1 Zelmanov had earlier made fundamental contributions to Jordan algebras and was an exp ert in this area, thus he was uniquely quali ed to attackthe restricted Burnside problem. In the sequel, the background from the theory of Jordan algebras and some of Zelmanov's contributions to this theory are rst discussed. Then the Burnside problems are describ ed and some of the things that were ear- lier known ab out them are listed. Section 4 contains some consequences of the restricted Burnside problem. Finally, some relevant results from Lie and Jordan algebras are mentioned. 1. Jordan Algebras Jordan algebras were intro duced in the 1930's by the physicist P. Jordan in an attempt to nd an algebraic setting for quantum mechanics. Hermitian matrices or op erators are not closed under the asso ciative pro duct xy ,but n are closed under the symmetric pro ducts xy + yx;xyx;x . An empirical investigation indicated that the basic op eration was the Jordan pro duct 1 x y = (xy + yx); 2 and that all other prop erties owed from the commutativelaw x y = y 2 2 x and the Jordan identity(x y ) x = x (y x). Jordan to ok these as axioms for the variety of Jordan algebras. Algebras resulting from the Jordan pro duct in an asso ciative algebra were called sp ecial. In a fundamental pap er Jordan, von Neumann, and Wigner classi ed all nite-dimensional formally-real Jordan algebras. These are direct sums of 5 typ es of simple algebras: algebras determined by a quadratic form on a vector space and algebras of hermitian n n matrices over the reals, complexes, quaternions, and o ctonions. The algebra of hermitian matrices over the o ctonions is Jordan only for n 3, and is exceptional (= non-sp ecial) if n = 3, so there was only one exceptional simple algebra in their list (now known as the Alb ert algebra of dimension 27). Algebraists develop ed a rich structure theory of Jordan algebras over elds of characteristic 6= 2. First, the analogue of Wedderburn's theory of nite dimensional asso ciative algebras was obtained by Alb ert. Next this was extended by Jacobson to an analogue of the Wedderburn-Artin theory of semisimple rings with minimum condition. The role of the one-sided ideals 2 was played by inner ideals, de ned as subspaces B of the algebra A such that U A B , where U y = 2x (x y ) (x x) y .If A is sp ecial, B x then U x = bxb in the asso ciative pro duct. Jacobson showed that every non- b degenerate Jordan algebra (U = 0 implies z = 0) with d.c.c. on inner ideals z is the direct sum of simple algebras which are of classical typ e. Up to this p oint, the structure theory treated only algebras with niten- ess conditions b ecause the primary to ol was the use of primitive idemp otents to intro duce co ordinates. In 1975 Alfsen, Schultz, and Stromer obtained a Gelfand-Neumark Theorem for Jordan C -algebras, and once again the basic structure theorem, but here again it was crucial that the hyp oteses guaran- teed a rich supply of idemp otents. In three pap ers (1979{1983), Zelmanovrevolutionized the structure theo- ry of Jordan algebras. These deal with prime Jordan algebras, where A is called prime if U C = 0 for ideals B and C in A, implies that either B or B C =0.Heproved the remarkable result that a prime non-degenerate Jordan algebra is either sp ecial of hermitian typeorofcli ordtyp e or is a form of the 27-dimensional exceptional algebra. The pro ofs required the intro duction of a host of novel concepts and techniques as well as sharp ening of earlier metho ds e.g. the co ordinatization theorem of Jacobson and analogues of the results on radicals due to Amitsur. One of the consequences of Zelmanov's theory is that the only exceptional simple Jordan algebras, even including in nite dimensional ones, are the forms of the 27-dimensional Alb ert algebras. Another consequence is that the free Jordan algebra in three or more generators has zero divisors (elements a so that U is not injective). This is in sharp contrast to the theorem of a Malcev and Neumann that any free asso ciative algebra can b e imb edded in a division algebra. Motivated by applications to analysis and di erential geometry,Koecher, Lo os, and Meyb erg extended the structure theory of Jordan algebras to tri- ple systems and Jordan pairs. Zelmanov applied his metho ds to obtain new results on these. To encompass characteristic 2 (which is essential for applications to the restricted burnside problem) it is necessary to deal with quadratic Jordan al- gebras. These were intro duced by McCrimmon and based on the axiomatized prop erties of the quadratic-linear pro duct U b. a In a joint pap er with McCrimmon, the results on prime algebras were extended to quadratic Jordan algebras. 3 2. Burnside Problems A group is local ly nite if every nite subset generates a nite group. In 1902 W. Burnside studied torsion groups and asked when such groups are lo cally nite. The most general form is the Generalized Burnside Problem. (GPB) Is a torsion group necessarily lo cally nite? e A group G has a nite exponent e if x =1forallx in G and e is the smallest natural numb er with this prop erty. A more restricted version of GPB is the ordinary Burnside Problem. (BP) Is every group which has a nite exp onent lo cally nite? There is a universal ob ject B (r;e), (the Burnside group of exp onent e on r generators) which is the quotient of the free group on r generators bythe th subgroup generated byalle powers. BP is equivalentto (BP)' Is B (r;e) nite for all natural numb ers e and r ? Burnside proved that groups of exp onent 2 (trivial) and exp onent 3 are lo cally nite. In 1905 Burnside showed that a subgroup of GL(n; C) of nite exp o- nent is nite. I. Schur in 1911 proved that a torsion subgroup of GL(n; C)is lo cally nite. This showed that the answers to BP or GBP would necessari- ly involve groups not discribable in terms of linear transformations over C. Other metho ds were required. During the 30's p eople b egan to study nite quotients of B (r;e)and considered the following statement. (RBP) B (r;e) has a unique maximal nite quotient RB(r,e). W. Magnus called the question of the truth or the falsityofRBP the restricted Burnside problem. If such a unique maximal nite quotient RB (r;e) exists for some r and e, then necessarily every nite group on r generators and exp onent e is a homomorphic image of RB (r;e). If RB (r;e) exists for some e and all r wesay that RBP is true for e. 3. Results In 1964 E. Golo d constructed in nite groups for every prime p, which are generated by 2 elements and in whichevery element has order a p ower of p, thus giving a negative answer to GBP. In 1968 S.I. Adian and P.S. Novikov showed that B (2;e) is in nite for e odd and e>4380, thus giving a negative answer to BP. In a seminal pap er P.