LIE RING METHODS IN THE THEORY OF FINITE NILPOTENT GROUPS

Transcription

1 307 LIE RING METHODS IN THE THEORY OF FINITE NILPOTENT GROUPS By GRAHAM HIGMAN 1. Introduction There are, of course, many connections between Group Theory and the theory of Lie rings, and my title is, perhaps, designed rather to exclude those I shall not deal with, than to define those that I shall. It excludes, for instance, the classical connection between Lie algebras and continuous groups, the applications, due to Mal'cev and others, to torsion-free nilpotent groups, and the construction, by Chevalley and Tits, using Lie algebra methods, of finite analogues to the exceptional simple Lie groups. There is one other connection, that could hardly have been excluded by title, but which I can no more than mention. This is the so-called Baker-Hausdorff formula, the theorem that, if x, y are non-commuting variables, the terms of the series log(e x.ey) = x + y + %[x,y]+^([[y,x],x] + [[x,y],y] +... are rational combinations of the elements of the Lie ring generated by x and y under the bracket multiplication [x,y] = xy yx. Under suitable restrictions (cf. Lazard E5] ) this fact leads to an elegant and precise correspondence between groups and Lie rings. For the sort of thing I have in mind, however, the restrictions are altogether too severe for the method to be of use. They are, in fact, precisely the sort ofthing one wants to have in the conclusion of one's theorems, rather than in the hypothesis. The method I want to discuss is much less precise, but much more generally applicable. It rests on the observation that the formal properties of multiplication and commutation in a group are similar to those of addition and multiplication in a Lie ring. For instance, commutation is almost bilinear: r n 1r _ r _ [xy,z] = y- 1 [x,z]y[y,z]; and, corresponding to the Jacobi identity, the product [I>, y},*] Ulf 9 z] 9 x\ [[z, x],y], if not 1, is at least expressible in terms of still more complicated commutators. These facts can be used to associate with a group O a Lie ring Lin the following manner.

2 308 GRAHAM HIGMAN First, one chooses a series of normal subgroups of G G = H 1^H 2^H Z =>...^H^... with the property that, for all integers i, j, the commutator group [Hi, Hj] is contained in H i+ j. For instance, the lower central series will do. Then the additive group of L is the direct sum of the (abelian) factor groups H^H^ of the series, elements of H^H^ being called homogeneous of degree i. Multiplication between homogeneous elements is defined by TJ IT r i IT 9% H i+i 9j H j+i = w> 9j\ üi+j+v That this is indeed well-defined, and that it can be extended by bilinearity to a multiplication on the whole of L, under which Xisa Lie ring, follows from the commutator identities already cited, and others like them. Obviously, this process can give information about G itself, rather 00 than about a factor group, only if f)h i = I, which implies that G is at worst w-nilpotent. If, however, this is so, then we can associate one or more Lie rings L with G, and, generally speaking, reasonable properties of G will translate into reasonable properties of L. Since the structure of L is richer, and in some ways more regular, than that of G, it is reasonable to hope that this translation will make problems easier to solve. I want to illustrate this by discussing two problems, both of which have their origins in the first decade of this century the golden age of group theory, if ever there was one. 2. Frobenius groups A Frobenius group is a permutation group on a finite set, no element of which, except the identity, has more than one fixed point. By a wellknown theorem of Frobenius, in such a group the elements without fixed points, together with the identity, form a normal subgroup G. But this by no means exhausts what can be said. If H is the subgroup leaving a particular point fixed, then of course H acts as a group of automorphisms of G, and it is easy to see that H acts regularly, that is, no element of H except the identity leaves fixed any element of G except the identity. The problem then is, what finite groups H act as regular automorphism groups, and what finite groups G admit regular automorphism groups? The answer to the first question, though a bit complicated, is classical. The investigation was begun by Burnside, and his work was

3 LIE RING METHODS 309 completed and corrected by Zassenhaus. But less is known about the answer to the second question. It is conjectured that G is nüpotent, but all that has been proved is that, if not, there must exist a simple G whose Sylow subgroups are large and possess improbable properties. Evidently, if G has a (finite) regular automorphism group, it has a regular automorphism of order p, for some prime p, so one may particularise the problem, and inquire, for each fixed prime p, what this implies about the structure of G. It is convenient at the same time to drop the assumption that G is finite, though to get a reasonable theorem we must have some condition on G, since a free product of p isomorphic groups certainly has a regular automorphism of order p. However, Neumann [6» 73 has shown that, under appropriate side conditions, a group with a regular automorphism of order p is abelian if p = 2, and nilpotent of class at most two if p = 3. It is in attempting to generalize this result that Lie ring methods come in. They cannot be used to prove G nilpotent. They can be used C3J to prove that if G is nilpotent, and has a regular automorphism of order p, there is a bound k(p) for its class, depending only on p. Naturally, there are two stages in the application. First one must translate the condition on G into a condition on L, and then one must prove a theorem on Lie rings satisfying this condition. The first half works precisely as one would expect. A nüpotent group G with a regular automorphism of order p has an associated Lie ring L of the same class with an induced automorphism which is also regular and of order p. Two warnings are, however, necessary. This result does not extend to w-nilpotent groups; and, if G is infinite, it may not be possible to take L to be the associated Lie ring formed from the lower central series itself. The second stage in the application is to prove that there is a function k(p) of p alone such that a Lie ring L with a regular automorphism of order p is nilpotent of class at most k(p). This depends on a rather complicated combinatorial argument which it is not possible even to sketch here, so that I must confine myself instead to making one or two remarks about it. First, an important step is to embed the given ring in one having as domain of operators the integer ring of the field of the jpth roots of unity, and generated by elements on which the automorphism acts as a scalar multiplier. It would be difficult, if not impossible, to carry out this step within G, and it is in this sense that the richer structure of L makes it easier to handle. Secondly, a much simpler argument proves an analogous theorem for

4 310 GRAHAM HIGMAN associative rings, and yields the value k(p) = p, which is easily seen to be best possible. But in the Lie ring case, apart from the first few values (k(2) = 1, Jfe(S) = 2, Jfc(6) = 6) au that is known is that k(p) ^ l(p 2 -l). Note, finally, that the Lie ring theorem is clearcut; the messy side conditions in the group result come in in the translation. 3. Burnside's problem The second application is, of course, to Burnside's problem, the problem, that is, of what can be said about a group G which satisfies an identical relation x n = 1, and, in particular, whether, if G is finitely generated, it is necessarily finite. Since our method requires that G is w-nilpotent, it will apply only to the case of a prime-power exponent, n = p r, say. Furthermore, it will apply only to the restricted Burnside problem, that is, to the question whether, for given k, there is a (finite) upper bound to the orders of the ^-generator groups of exponent n. As before, there are two stages in the application of the method. The first requires us to translate the group theoretical problem into Lie ring terms; the second to prove a theorem about Lie rings. It cannot be said as yet that we know how to accomplish either half of this programme, in general. Let us look at the first stage first. If G is an w-nilpotent group of exponent n = p r, what can be said about its associated Lie ring Li We shall assume that L is formed using the lower central series of G; this has the advantage that if G can be generated by k elements, so can L; which is obviously desirable in the present context. Evidently, L is of characteristic n; that is, nx = 0 for all x in L. But, furthermore, L satisfies certain identical relations. Let us explain how these can be obtained. We begin with a word WQ X-^XQ Xf in the free group on generators a v..., a N, which is a product oînth powers ; and for simplicity we suppose that w 0 is a positive word, so that each x i is an dj. We apply to w 0 a commutator collection process. In the first stage of this the letters other than a x are collected to the left, in the order in which they occur. In the process, evidently, we shall have to introduce commutators [%, a { ], and then [a v a i9 a^], and so on. The upshot is an expression in which each y i is an a p j #= 1, and each z i is a x, or a commutator con-

5 LIE RING METHODS 311 taining a v Evidently, y x y 2... y s is the result of putting % = 1 in w Q, so that it is a product of nth powers. Hence w 1 = z 1 z 2...z u is a product of nth powers. In the second stage, we similarly collect to the left, and drop, those z t which do not involve a 2. The result is a product w 2 of commutators, all involving both a ± and a 2, which is also a product of nth powers. Continuing thus, we ultimately obtain a product w N of commutators, all involving each of a v..., a N, which is also a product of nth powers. Now a commutator of weight N in w N involves a ± just once, and hence was introduced at the first stage of the collection process. Hence it is of the form [a x,a 2(r,...,a N(T ], where cr is a permutation of 2,..., N; and it is easy to see that the number k a of times this commutator occurs is equal to the number of times a v a 2(r,...,#JVO- occurs as a subsequencein Wo. Thus UKh.<h~-,**.?' er is a product of nth powers, and commutators of weight greater than N involving each e^. It is easy to see that 2 #cr^l^2o- X Na = 0 cr is an identical relation in L. If, in particular, we take w Q = (a x a 2... a n ) n, this relation becomes From this point, it is necessary to distinguish the case n p from the general case n = p r, r > 1. If n = p, the condition we have obtained is precisely the (p l)st Engel condition, xyv- 1 = 0. That is, the associated Lie ring of a group of exponent p is of characteristic p and satisfies the (p l)st Engel condition. As far as I know, we cannot be sure, except for the smallest values of p, that this gives the complete translation of the problem. But, what is far more important, we do now know that it is sufficient. That is, a Lie ring of characteristic p satisfying the (p l)st Engel condition is locally finite. For p = 2 this is obvious; for p = 3 it was proved essentially by Levi and van der Waerden, though their argument was conducted entirely in the group. The case p = 5 is due independently to Kostrikin (see references in [4] ) and to me C2]. The general case has been obtained very recently indeed by Kostrikin [4], whose work must be considered the first major break-through in Burnside's problem. It is not, of course, possible here to give any details, though it is perhaps worth observing that the point here is that the behaviour of

6 312 GRAHAM HIGMAN L is more regular than that of G. It is not obvious, for instance, that G is an Engel group; certainly it is not true, in general, that for x, y in G, [x,y,...,y] = 1. p-i For q = p r (r > 1), the position is much less satisfactory. We know, in fact, that the translation mentioned above (which is now not a genuine Engel condition, but only a linearized one) is not the whole story, nor is it sufficient to prove L locally finite (e.g. the relation obtained lies in the second derived ring (L 2 ) 2, whereas for large enough k, xy k = 0; also, for g = 4, the relation holds in the non-soluble ring of dimension 3). In view of Kostrikin's achievement, it is, perhaps, appropriate to conclude with a word or two about the present state of Burnside's problem. It has for some time been clear that the problem falls into four parts: (i) the nilpotent part, the restricted problem for (a) prime and (6) prime power exponents; (ii) the reduction to (i) of the problem as far as it concerns soluble groups (this deals, in particular, with exponents involving just two primes); (iii) the reduction to (i) and (ii) of the general restricted problem; (iv) the passage from the restricted problem to the general problem. Of these parts, (ii) has been dealt with completely in Hall and Higman [1], and Kostrikin has well begun (i). It seems likely that primepowers will fall to methods of the same general nature. There remain (iii) and (iv). Of these, (iv) is, perhaps, the most characteristic of the problem; (iii) draws attention to our need for information about the finite simple groups, but it scarcely needs Burnside's problem to do that. REFERENCES [1] Hall, P. and Higman, G. On the p-lengths of ^-soluble groups, and reduction theorems in Burnside's problem. Proc. Lond. Math. Soc. (3), 6,1-42 (1956). [2] Higman, G. On finite groups of exponent five. Proc. Camb. Phil. Soc. 52, (1956). [3] Higman, G. Groups and rings having automorphisms without non-trivial fixed elements. J. Lond. Math. Soc. 32, (1957). [4] KOCTPHKHH, A. H. O npoßjieme 6epHcaü;n;a. ffonjiadu AnadeMuuHayn CCCP, N.S., 119, (1958). [5] Lazard, M. Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Éc.Norm. Sup. (3), 71, (1954). [6] Neumann, B. H. On the eommutativity of addition. J. Lond. Math. Soc. 15, (1940). [7] Neumann, B. H. Groups with automorphisms that leave only the neutral element fixed. Arch. Math. 7, 1-5 (1956).