On the Work of Efim Zelmanov
WALTER
FEIT
Department of Mathematics, Yale University
New Haven, CT 06511, USA
0 Introduction
Efim Zelmanov has received a Fields Medal for the solution of the restricted Burnside problem. This problem-in group theory-had long been known-to be-relatedto the theory of Lie algebras. In fact, to a large extent it is the problem in Lie
algebras. A precise statement of it can be found in Section 2 below.
In proving the necessary properties of Lie algebras, Zelmanov built on the
work of many others, though he went far beyond what had previously been done
in this direction. For instance, he greatly simplified Kostrikin's results [K] which
settled the case of prime exponent and then extended these methods to handle the
prime power case.
However, while the case of exponent 2 is trivial, the case of exponent 2k for
arbitrary h is the most difficult case that needed to be addressed. The results from
Lie algebras that work for exponent ph with p an odd prime are not adequate for
exponent 2fc. This indicated that a new approach was necessary here. Zelmanov
was the first to realize that in the case of groups of exponent 2k the theory of
Jordan algebras is of great significance. Even though Vaughan-Lee later removed
the need for Jordan algebras [V], it seems probable that this proof could not have
been discovered without them, as the ideas used arise most naturally from Jordan
algebras.
Zelmanov had earlier made fundamental contributions to Jordan algebras and
was an expert in this area, thus he was uniquely qualified to attack the restricted
Burnside problem.
Below the background from the theory of Jordan algebras and some of Zelmanov's contributions to this theory are first discussed (I am grateful to McCrimmon and Jacobson for much of this material). See [Jl] and [J2] for the general
theory of Jordan algebras. Then the Burnside problems are described and some of
the things that were earlier known about them are listed. Section 4 contains some
consequences of the restricted Burnside problem. Finally, some relevant results
from Lie and Jordan algebras are mentioned (in a necessarily sketchy manner).
Zelmanov himself has written a set of expository notes on these topics [ZÌI]. It
contains all the appropriate definitions and some of the material used in the proof
of the restricted Burnside problem. It also includes material on several related questions, such as the Kurosh-Levitzky problem. Of course, ultimately, the details are
the heart of the matter, and for these the reader should consult [Z8] and [Z9], or [V].
Proceedings of the International Congress
of Mathematicians, Zürich, Switzerland 1994
© Birkhäuser Verlag, Basel, Switzerland 1995
18
Walter Feit
This circle of ideas illustrates the unity of mathematics once again. Although
many formal identities are used to settle the restricted Burnside problem, it seems
unlikely that they could have been discovered without the conceptual framework
provided by the seemingly unrelated and diverse fields of Lie and Jordan algebras.
1 Jordan algebras
Jordan algebras were introduced in the 1930s by the physicist P. Jordan in an
attempt to find an algebraic setting for quantum mechanics, essentially different
from the standard setting of hermitian matrices. Hermitian matrices or operators
are not closed under the associative product xy, but are closed under the symmetric
products xy + yx, xyx, xn. An empirical investigation indicated that the basic
operation was the Jordan product
x-y=
-(xy + yx),
and that all other properties flowed from the commutative law x-y = y-x and the
Jordan identity (x2 • y) • x = x2 • (y • x). (For example, the Jordan triple product
{xyz} = ^ (xyz + zyx) can be expressed as x • (y • z) + (x • y) • z — (x • z) • y, though
the tetrad
{xyzw} = -(xyzw + wzyx)
z
cannot be expressed in terms of the Jordan product.) Jordan took these as axioms
for the variety of Jordan algebras. Algebras resulting from the Jordan product in
an associative algebra were called special, so the physicists were seeking algebras
that were exceptional (= nonspecial). In a fundamental paper [JNW] Jordan, von
Neumann, and Wigner classified all finite-dimensional formally-real Jordan algebras. These are direct sums of five types of simple algebras: algebras determined by
a quadratic form on a vector space (a special subalgebra of the Clifford algebra of
the quadratic form) and four types of algebras of hermitian (n x n)-matrices over
the four composition algebras (the reals, complexes, quaternions, and octonions).
The algebra of hermitian matrices over the octonions is Jordan only for n < 3,
and is exceptional if n = 3, so there was only one exceptional simple algebra in
their list (now known as the Albert algebra, of dimension 27). At the end of their
paper Jordan, von Neumann, and Wigner expressed the hope that by dropping the
assumption of finite dimensionality one might obtain exceptional simple algebras
other than Albert algebras.
Algebraists developed a rich structure theory of Jordan algebras over fields of
characteristic ^ 2. First, the analogue of Wedderburn's theory of finite-dimensional
associative algebras was obtained by Albert. Next this was extended by Jacobson
to an analogue of the Wedderburn-Artin theory of semisimple rings with minimum
condition on left or right ideals. In this, the role of the one-sided ideals was played
by inner ideals, defined as subspaces B such that Ubx is in B for all x in the
algebra A and all b in B where Ua = 2L2a — Lai and La is the left multiplication by
a in the Jordan algebra A. If A is the Jordan subalgebra of an associative algebra
then Ubx = bxb in the associative product. Using the definition of semi-simplicity
On the Work of Efim Zelmanov
19
(called nondegeneracy) that A contains no z ^ 0 such that Uz = 0, Jacobson
showed that every nondegenerate Jordan algebra with d.c.c. on inner ideals is the
direct sum of simple algebras that are of classical type (analogues of those found
in [JNW]: (Type I) Jordan algebras of nondegenerate quadratic forms; (Type II)
algebras II(A, *) of hermitian elements in a *-simple artinian associative algebra
A ((nx n)-matrices over a division algebra with involution, or over a direct sum of
a division algebra and its opposite under the exchange involution, or matrices over
a split quaternion algebra with standard involution); (Type III) 27-dimensional
exceptional Albert algebras; (Type IV) Jordan division algebras, defined by the
condition that Ua is invertible for eveiy a ^ 0.
Up to this point, the structure theory treated only algebras with finiteness
conditions because the primary tool was the use of primitive idempotents to introduce coordinates^ In 1975 Alfsen, Schultz, and Stonier obtained a Gelfand-Naimark
theorem for Jordan C*-algebras, and once again the basic dimensional structure
theorem, but here again it was crucial that the hypotheses guaranteed a rich supply
of idempotents.
In three papers [Zl], [Z2], [Z3], Zelmanov revolutionized the structure theory
of Jordan algebras. These deal with prime Jordan algebras, where A is called prime
if UBC = 0 for ideals B and C in A implies that either B or C = 0. In [Zl] Zelmanov proved the remarkable result that a prime Jordan algebra without nil ideals
(improved in [Z3] to prime and nondegenerate) is either i-special (a homomorphic
image of a special Jordan algebra) or is a form of the 27-dimensional exceptional
algebra. This applied in particular to simple algebras. The proof required the introduction of a host of novel concepts and techniques as well as sharpening of
earlier methods, e.g. the coordinatization theorem of Jacobson and analogues of
results on radicals due to Amitsur.
The paper [Z3] is devoted to the study of z-special Jordan algebras. Zelmanov
showed that a prime nondegenerate i-special algebra is special, and he determined
their structure as either of hermitian type or of Clifford type. Paper [Z2], which
preceded [Z3] obtained these results for Jordan division algebras.
The principal tool in both papers is the study of the free associative algebra
®{X) on X = {x1,x2,...}.
This becomes a Jordan algebra $(X)+ by replacing the
given associative multiplication ab by a • b = ^(ab-\-ba). The subalgebra SJ(X) of
<&(X)+ generated by X is called the free special Jordan algebra.
We also have the subalgebra II(X) of $(X)+ of symmetric elements (a" = a)
under the involution in ®(X) fixing the elements of X. It was shown by Paul Cohn
in 1954 that SJ(X) C H(X) and H(X) is the subalgebra of $(X)+ generated by
X and all the tetrads x%x3xhxl with i < j < k < I. Zelmanov has obtained a
completely unanticipated supplement to Cohn's theorem: thé existence of elements
/ in SJ(X) such that if / ( / ) denotes the ideal generated by / then
{I{f),p,q,r}£SJ(X)
for p, q, r in SJ(X). This is used to sort out the two types of z-special algebras:
Clifford types characterized by the identity / = 0 and hermitian types by the
nonidentity / ^ 0.
20
Walter Feit
One of the consequences of Zelmanov's theory is that the only exceptional
simple Jordan algebras, even including infinite-dimensional ones, are the forms
of the 27-dimensional Albert algebras. This laid to rest the hope that had been
raised by Jordan, von Neumann, and Wigner in [JNW]. Another consequence of
Zelmanov's results is that the free Jordan algebra in three or more generators has
zero divisors (elements a such that Ua is not injective). This is in sharp contrast
to the theorem of Malcev and Neumann that any free associative algebra can be
imbedded in a division algebra.
Motivated by applications to analysis and differential geometry, Koecher,
Loos, and Myberg extended the structure theory of Jordan algebras to triple systems and Jordan pairs. Zelmanov applied his methods to obtain new results on
these.
Lie methods were used in these papers based on the Tits-Koecher construction. The final work in this line of investigation was [Z4] in which Zelmanov applied
the theory of Jordan triple systems to study graded Lie algebras with finite gradings in which the homogeneous parts could be infinite dimensional.
To encompass characteristic 2 (which is essential for applications to the restricted Burnside problem) it is necessary to deal with quadratic Jordan algebras
[JM]. These were introduced by McCrimmon in [Mc] as the natural extension of
Jordan algebras to algebras over any commutative ring. This amounted to replacing the product a- b = ^(ab + ba) in an associative algebra by the product Uab.
In the joint paper with McCrimmon [ZM], the results of [Z3] were extended
to quadratic Jordan algebras.
2 Burnside problems
We begin with some definitions and notation.
A group is locally finite if every finite subset generates a finite group. In
1902 Burnside [Bl] studied torsion groups and asked when such groups are locally
finite. The most general form of the question is the Generalized Burnside Problem
(GBP).
(GBP) Is a torsion group necessarily locally finite?
Equivalently
(GBP)' Is every finitely generated torsion group finite?
A group G has a finite exponent e if xe = 1 for all x in G and e is the
smallest natural number with this property. Clearly a group with a finite exponent
is a torsion group. A more restricted version of GBP, which already occurs in
Burnside's work, is the ordinary Burnside Problem (BP).
(BP) Is every group that has a finite exponent locally finite?
There is a universal object B(r,e), (the Burnside group of exponent e on r
generators), which is the quotient of the free group on r generators by the subgroup
generated by all eth powers. BP is equivalent to
(BP)' Is B(r, e) finite for aU natural numbers e and r?
On the Work of Efim Zelmanov
21
Burnside proved that groups of exponent 2 (trivial) and exponent 3 are locally
finite. In 1905 Burnside [B2] showed that a subgroup of GL(n, C) of finite exponent
is finite. Schur in 1911 [Sc] proved that a finitely generated torsion subgroup of
GL(n, C) has finite exponent, and hence a torsion subgroup of GL(n, C) is locally
finite. This was very important as it showed that answers to BP or GBP would
necessarily involve groups not describable in terms of linear transformations over C.
Other methods were required. In handling groups of exponent 3 Burnside had
used only the multiplication table of a group. However, his methods were totally
inadequate to handle, for instance, groups of prime exponent greater than 3.
During the 1930s people began to study finite quotients of B(r,e) and considered the following statement.
(RBP) B(r,e) has only finitely many finite quotients.
This is equivalent to"
(RBP)' B(r,e) has a unique maximal finite quotient
W. Magnus called the question
Buniside problem. If such a unique
some e and 7', then necessarily every
is a homomorphic image of RB(r, e).
that RBP is true for e.
RB(r,e).
of the truth or falsity of RBP the restricted
maximal finite quotient RB(r,e) exists for
finite group on r generators and exponent e
If RB(r, e) exists for some e and all r we say
3 Results
In 1964 Golod [G] constructed infinite groups for every prime p, which are generated by 2 elements and in which every element has order a power of p, thus
giving a negative answer to GBP. A few years later in 1968 Adian and Novikov
[AN] showed that B(2,e) is infinite for e odd and e > 4380, thus giving a negative
answer to BP. The bound has been improved since then as B(r,e) is finite for
e = 2,3,4, or 6, but in no other case with r > 1 is it known to be finite.
In a seminal paper Hall and Higman [HH] in 1956 proved a series of results
concerning RBP. Let n be a set of primes. Consider the following two statements.
(1)
(2)
There are only finitely many finite simple 7r-groups of any given exponent.
The Schreier conjecture is true for 7r-groups, i.e. for any finite simple 7r-group
G, Aut(G)/G is solvable.
A special case of one of their results is the following.
[HH], Suppose that statements (1) and (2) are true for the set TT. Then
if for every prime p in n, and natural numbers m and r} RB(r,pm) exists; then
RBP is true for any exponent e that is a ir-number.
THEOREM
The classification of the finite simple groups shows that (1) and (2) are true
for any set of primes IT. Hence the truth of RBP will follow once it is proved that
RB(r,pm) exists for all primes p and all natural numbers 777, and r.
In 1959 Kostrikin announced that RB(r,p) exists for p a prime and any
natural number r. Kostrikin's original argument had some difficulties. He published a corrected and updated version of his proof in his book [K], which contains
numerous references to Zelmanov.
22
Walter Feit
In 1989 Zelmanov announced that RBP is true for all exponents pm with p
any prime, and hence for aU exponents by the remarks above. The proof appeared
in 1990-91 in Russian. The English translation appeared in [Z8] and [Z9].
It should be mentioned that analogous questions have been raised for associative, Lie, and Jordan algebras. Golod's work was actually motivated by the
associative algebra question and the counterexamples for groups arose as corollaries. The questions for Lie and Jordan algebras will be discussed below.
4 Some consequences
This section contains some consequences of RBP. The ideas used in the proof, in
addition to the actual result, have also been applied widely.
The next three results were proved by Zelmanov [ZIO] as direct consequences
of RBP.
THEOREM
1. Every periodic pro-p-group is locally finite.
2. Every infinite compact (Hausdorff) group contains an infinite
abelian subgroup.
COROLLARY
THEOREM
3. Every periodic compact (Hausdorff) group is locally finite.
Theorem 3 was conjectured by Platonov [Ko].
Shalev showed that RBP implies the following.
4 [Sh]. A pro-p-group is p-adic analytic if and only if there exists a
natural number n such that the wreath product Zp I Zpn is not a homomorphic
image of any subgroup of G.
THEOREM
The "only if" part of Theorem 4 is elementary, but the converse is equivalent to RBP. Since then, Zelmanov jointly with others, has made several further
contributions to the study of pro-p-groups, see e.g. [ZS], [ZW].
5 Lie algebras
Let G be a finite group of exponent pk, p a prime. Let G — GQ and Gi+1 — [G, GJ
for all i. Choose s with G s / (1), G a + 1 = (1). Then
G = G0 > • • • > G s+1 = (1)
is the lower central series of G. Define
L(G) =
£G./G.
+ 1
4= 0
as abelian groups. Then L(G) becomes a Lie ring with [atGti a3G3] = [ati a3]Gl+3+1,
and L(G) has the same nilpotency class as G. Furthermore L(G)jpL(G) is a Lie
algebra over Zp.
Let L be a Lie algebra.
L satisfies the Engel identity (En) if ad(x)n = 0 for all x in L.
An element x in L is nilpotent if ad(x)n = 0 for some n.
If G has exponent p then L(G) is a Lie algebra over Zp that satisfies (Ep_1).
Kostrikin proved
On the Work of Efim Zelmanov
23
5 [K]. If L is a Lie algebra over Zp that satisfies (Ep_1) then L is
locally nilpotent.
THEOREM
Theorem 1 implies the existence of RB(r,p) and so yields RBP for prime
exponent. Observe that for prime exponent e = p, the case p = 2 is trivial, so that
it may be assumed that p > 2. This is in sharp contrast to prime power exponents
e = pk, where p = 2 is the most complicated case.
An element a of L is a sandwich if [[L, a], a] = 0 and [[[L, o],L], a] = 0. L is a
sandwich algebra if it is generated bjr finitely many sandwiches. This concept was
introduced by Kostrikin and is of fundamental importance for the proof of RBP.
A first critical result is
THEOREM
6 [ZK]. Every sandwich Lie algebra is locally nilpotent.
Theorem 6 is essential for the proof of Theorem 5.
The main result in [Z8] is rather technical but it has the following consequence.
THEOREM
7 [Z8]. Every Lie ring satisfying an Engel condition is locally nilpotent.
More importantly, it implies
THEOREM 8. RB(r,pk)
exists for p an odd prime.
Once again an essential part of the proof requires Theorem 2. Let L be a Lie
algebra over an infinite field of characteristic p that satisfies an Engel condition.
The way to apply Theorem 2 is to construct a polynomial f(x1,...,xt)
that is
not identically zero, such that every element in f(L) is a sandwich in L. Actually
such a polynomial is not constructed but its existence for p > 2 follows only after
a very complicated series of arguments, which constitute the bulk of the paper
[Z8]. This of course settles RBP for odd exponent. (It might be mentioned that
the classification of finite simple groups is not required here, only that groups of
odd order are solvable.)
6 The case of exponent 2fc
The outline of the proof of RBP for exponent 2k is similar to that for exponent
pk with p > 2 described in the previous section. However, the construction of
the function / is vastly more complicated. It is here that quadratic Jordan algebras play an essential role, most especially the results of [ZM]. The details are
extremely technical and cannot be presented here. The reader should consult [Z9]
for a complete proof.
General References
[AN]
[Bl]
S. I. Adian and P. S. Novikov, On infinite periodic groups, I, II, III, Math.
USSR-Izv. 2 (1968), 209-236, 241-479, 665-685.
W. Burnside, On an unsettled question in the theory of discontinuous groups,
Quart. J. Math. 33 (1902), 230-238.
24
Walter Feit
[B2]
, On criteria for the finiteness of the order of a group of linear substitutions,
Proc. London Math. Soc. (2) 3 (1905), 435-440.
[G]
E. Golod, On nil algebras and residually finite p-groups, Math. USSR-Izv. 28
(1964), 273-276.
[HH]
P. Hall and G. Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. 6 (1956), 1-40.
[Jl]
N. Jacobson, Structure and representation of Jordan algebras, Amer. Math. Soc,
Providence, RI 1968.
[J2]
, Structure theory of Jordan algebras, Univ. of Arkansas Lecture Notes 5
(1981).
[JM]
N. Jacobson and K. McCrimmon, Quadratic Jordan algebras of quadratic forms
with base points, J. Indian Math. Soc. 35 (1971), 1-45.
[JNW] P. Jordan, J. von Neumann, and F. Wigner, On an algebraic generalization of
the quantum mechanical formalism, Ann. of Math. (2) 35 (1934), 29-73.
[K]
A. I. Kostrikin, Around Burnside, (Russian) MR (89d,20032).
[Ko]
Kourovka Notebook, 12th ed., Inst. Mat. SO AN SSSR Novosibirsk, 1992. Problem 3.41.
[Mc]
K. McCrimmon, A general theory of Jordan rings, PNAS 56 (1966), 1072-1079.
[Sc]
I Schur, Über Gruppen periodischer Substitutionen, Collected Works I, 442-450,
Springer, Berlin, Heidelberg, and New York 1973.
[Sh]
A. Shalev, Characterization ofp-adic analytic groups in terms of wreath products,
J. Algebra 145 (1992), 204-208.
[V]
M. R. Vaughan-Lee, The restricted Burnside problem, 2nd ed., Oxford University
Press, Oxford, 1993.
References to Zelmanov's Work
[ZI]
[Z2]
[Z3]
[Z4]
[Z5]
[Z6]
[Z7]
[Z8]
[Z9]
[ZIO]
[ZÌI]
[ZK]
[ZM]
[ZS]
[ZW]
E. Zelmanov, Primary Jordan algebras, Algebra and Logic 18 (1979), 103-111.
, Jordan division algebras, Algebra and Logic 18 (1979), 175-190.
——, On prime Jordan algebras II, Siberian Math. J. 24 (1983), 73-85.
, Lie algebras with a finite grading, Math. USSR-Sb. 52 (1985), 347-385.
, On Engel Lie Algebras, Soviet Math. Dokl. 35 (1987), 44-47.
, Engelian Lie Algebras, Siberian Math. J. 29 (1989), 777-781.
, Weakened Burnside problem, Siberian Math. J. 30 (1989), 885-891.
, Solution of the restricted Burnside problem for groups of odd exponent,
Math. USSR-Izv. 36 (1991), 41-60.
, A solution of the restricted Burnside problem for 2-groups, Math. USSRSb. 72 (1992), 543-564.
, On periodic compact groups, Israel J. Math. 77 (1992), 83-95.
, Nil rings and periodic groups, Korean Math. Soc. Lecture Notes 1992,
Korean Math. Soc. Seoul, South Korea.
and A. I. Kostrikin, A theorem on sandwich algebras, Proc. Steklov Inst.
Math. 183 issue 4 (1991), 121-126.
and K. McCrimmon, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), 113-122.
and Shalev, Pro-p-groups of finite coclass, Math. Proc. Cambridge Philos.
Soc. I l l (1992), 417-421.
and J. S. Wilson, Identities of Lie algebras for pro-p-groups, J. Pure Appi.
Algebra 81 (1992), 103-109.