CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS

Transcription

1 Actes, Congrès intern, math., Tome 1, p. 355 à 360. CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS by DANIEL GORENSTEIN It has been known for a long time that the structure of a finite simple group is intimately connected with the structure of the centralizers of its involutions. An old result of Brauer and Fowler asserts, in fact, that there are at most a finite number of simple groups in which the centralizer of an involution has a given structure. A more specific, pioneering result of Brauer established that the groups PSL(3, q) with q = l(mod 4) and the Mathieu group M u were the only simple groups in which the centralizer of an involution was isomorphic to a homomorphic image of GL(2, q) by a central subgroup of odd order. This last theorem was certainly one of the first of what has now become a major area of finite group theory, the characterization of the presently known simple groups in terms of the structure of the centralizers of their involutions. This work is developing at such a pace that it is not unreasonable to hope that within a very few years such characterizations will exist for all the known simple groups. We should mention that some of these investigations have led to the discovery of certain of the new sporadic simple groups. In fact, the first of these was discovered by Janko while studying groups in which the centralizer of an involution was isomorphic to the direct product of a group of order 2 and A 5. In all these theorems one specifies to begin with the structure of the centralizer of one or more involutions of an abstract simple group G and then tries to prove that the structure of G (that is, its multiplication table) is essentially uniquely determined in terms of a set of generators and relations by the given conditions. On the other hand, in more general classification problems the objective of the analysis is, in contrast, the determination of the structure of these centralizers in the group under investigation. Once this is accomplished, the problem is thus reduced to precisely the kind of characterization theorem just described. For example, in the study of simple groups whose Sylow 2-subgroups are either quasi-dihedral or a wreath product of a cyclic group of order at least 4 by a group of order 2, which have recently been completely characterized by Alperin, Brauer, and myself, almost all of our effort was devoted to establishing that in such a group G the centralizer of an involution is necessarily isomorphic to a homomorphic image of either GL(2, q) or G U(2, q) for some odd q by a central subgroup of odd order. Using the above-mentioned result of Brauer together with other related characterization theorems of Brauer, O'Nan, and Suzuki, we were able to conclude that G was isomorphic to one of the groups PSL(3, q), PSU(3, q), or M n. Even a cursory glance at the various general classification problems solved to date

2 356 D. GORENSTEIN B3 will reveal the essential role played by the centralizers of involutions in each of the proofs. It is therefore natural to raise the following general question: What can one say about the centralizers of involutions in arbitrary finite simple groups? Put in this form, the question is actually too general and probably unattackable, for it omits an essential ingredient of each of the successfully completed general classification problems : namely, the role played by induction. For example, in the quasidihedral and wreathed problem, our group G was by assumption a minimal counterexample to the desired classification theorem. But then using induction together with the previously obtained classification of groups with dihedral Sylow 2-subgroups together with the solvability of groups of odd order, we were able to determine the general shape of every proper subgroup of G. Without such knowledge, it would have been impossible for us to have carried out the so-called local group-theoretic analysis that constitutes the non character-theoretic portion of the proof. Moreover, it is precisely by means of extensions of this type of local group-theoretic analysis that the attack on the general problem posed above is to be made. We see then that some hypotheses on the proper subgroups of G must be imposed if we are to expect to obtain any reasonable answers to our question. The most natural general condition is clearly the following: the composition factors of every proper subgroup of G are among the presently known simple groups. Indeed, in any specific general classification problem a minimal counterexample will always be a simple group of this type. Of course, in each particular argument only certain properties of the known simple groups will actually be used. It turns out, in fact, that only very few of what appear to be general properties of the presently known simple groups enter into the analysis. These properties can actually be systematically formalized and, moreover, it is important to proceed formally since we want our results to remain valid, if at all possible, even if new simple groups are discovered in the future. It is not surprising that not every composition factor of every proper subgroup of our group G plays a role; in fact, only certain composition factors of the centralizers of the involutions of G are critical. To describe these, we introduce some terminology. A quasisimple group is by definition a perfect central extension of a simple group and a semisimple group is any central product of quasisimple groups. Observe that any group H possesses a unique largest normal semisimple subgroup. As usual, 0(H) denotes the unique largest normal subgroup of H of odd order, the so-called (2-regular) core of H. The key notion is that of L(G), which is a certain collection of quasisimple groups associated with the group G. A given quasisimple group L is in L(G) if and only if L is isomorphic to one of the quasisimple components of the largest normal semisimple subgroup of C G (x)/0(c G (x)) for some involution x of G. It is primarily properties of the elements of L(G) that are needed for our arguments. For a given group G, L(G) may, of course, be empty. Since each element of L(G) is nonsolvable, this will clearly be the case if the centralizer of every involution of G

3 CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS 357 is solvable. Actually L(G) is empty if and only if each such centralizer is 2-constrained. By definition, a group H is 2-constrained if C B (0 2 (H)) ç 0 2 (H), where H = H/0(H) and 0 2 (H) denotes the largest normal 2-subgroup of H. What kind of results about the centralizers of involutions would one hope to establish? Obviously we would like to prove that these centralizers resemble those in the presently known simple groups, the closer, the better. Let us then briefly examine the centralizers of involutions in these groups. Apart from the alternating groups and the sporadic groups (and, of course, those of prime order), all the remaining known groups are of Lie type. Moreover, for a group L of the latter type, the centralizer of an involution r of L has sharply divergent form according as L is defined over a field of even or odd characteristic. This is natural since t is correspondingly a unipotent or semisimple element of L. In the even characteristic case, it appears that H = C L (t) is always 2-constrained and has a trivial core. In particular, the largest normal semisimple subgroup K of H is trivial. By contrast, in the odd characteristic case, it appears to be generally true that H/K is a small solvable group and that K has 1 or 2 components which are of Lie type of odd characteristic (except in certain degenerate cases in which the number of components is 0, 3, or 4). Thus in the odd case, K dominates the structure of H. The centralizers of involutions in the sporadic groups have structures similar to those in the groups of Lie type of even characteristic period. (In some instances, K is non-trivial, but in such cases K/Z(K) is isomorphic to a group of Lie type of even characteristic). In A, the centralizers of involutions have features of those in groups of Lie type of both even and odd characteristic. I should like now to illustrate these considerations by describing two general results which pertain to the even and odd characteristic cases respectively. These results represent a joint effort with John Walter. We have seen above that 0(C L (t)) = 1 in the characteristic 2 case. Let us say that any quasisimple group L with this property is 1-balanced. In general, this property is false if L is of Lie type of odd characteristic or isomorphic to A with n = 3 (mod 4), but holds for the remaining known simple groups. A second condition which we need to state our first result is called 2-generation. A quasisimple group L is said to be 2-generated if for any Sylow 2-subgroup R of L, we have L = < N L (Q) Q ^ R, Q contains a noncyclic abelian subgroup >. (We have here simplified both these definitions slightly; actually it is necessary to impose the conditions on certain collections of groups which contain L as a normal subgroup). The so-called Bender groups PSL(2, 2"), Sz(2 n ) 9 and PSU(3 9 2 n ) and any of their central extensions by a group of order 2 are not 2-generated, as is easily checked, since in any of these groups a Sylow 2-subgroup is disjoint from its conjugates. Apart from the Bender groups, the only other known quasisimple groups that are not 2-generated are Janko's first group mentioned above and the perfect central extension  9 of A 9 by a group of order 2. For brevity, we call any one of the groups on this list exceptional. Finally a group G is said to have 2-rank or normal 2-rank at least k if a Sylow 2-subgroup of G possesses respectively an abelian or normal abelian subgroup of rank at

4 358 D. GORENSTEIN B3 least k. In particular, G has 2-rank 1 if and only if G possesses no non-cyclic abelian 2-subgroups and hence by a well-known result if and only if G has cyclic or generalized quaternian Sylow 2-subgroups. We can now state our first result. THEOREM. Let G be a simple group of normal 2-rank at least 3. If every element of L(G) is 1-balanced and either 2-generated or exceptional, then 0(C G (x)) = 1 for every involution x of G. Actually in this degree of generality, some technical additional assumptions must be made. However, the stated result does hold if L(G) is empty and hence if the centralizer of every involution of G is 2-constrained. Likewise it holds if every element of L(G) is 2-generated. This result, although very powerful in certain situations, still leaves one, in general, a long way from pinning down the structure of the centralizers of the involutions in such a group G, which we may view as the general group of " characteristic 2 " type. The central problem in the classification of simple groups of characteristic 2 type (which we note includes all the presently known sporadic groups along with the groups of Lie type of characteristic 2) is the development of general methods which will enable one to restrict the structure of these centralizers much more sharply. A major portion of Thompson's celebrated N-group paper (sections 8, 9, 13, 14, and 15) deals with a particular case of this problem. It will be important in this connection to determine how far his methods and results can be extended. In contrast, our results in the odd characteristic case are already quite definitive. We shall not attempt to state here the exact set of conditions which we impose on the elements of L(G), as some are fairly technical. Again they involve certain notions of balance and generation. They are embodied in the concepts of what we call a A-group and a weak A-group. The central point about a A-group or weak A-group G is that, in effect, we assume that the elements of L(G) are of known type with at least one of these elements (but not necessarily all) being a group of Lie type of odd characteristic (and not isomorphic to one of even characteristic). The sole distinction between a A-group and a weak A-group is that in the former case the groups A n and  n with n 3 divisible by a high power of 2 are excluded from L(G). These particular groups require special treatment in our analysis, being the only known groups which do not have the property of what we call 3-balance. To state our principal result in the odd characteristic, we need one further notion. A group H is said to have standard form if H possesses a normal quasisimple subgroup L such that C H (L) has cyclic or generalized quaternion Sylow 2-subgrqups. L is called the standard component of H. Note that the possible structures of C H (L) are very restricted in this case and are completely known. Moreover, H/LC H (L) is isomorphic to a group of outer automorphisms of L. Hence the structure of H is essentially completely determined once the standard component L of H is specified. Our main result asserts THEOREM. If G is either a simple A-group of normal 2-rank at least 13 or a simple

5 CENTRALIZERS OF INVOLUTIONS IN FINITE SIMPLE GROUPS 359 weak A-group of normal 2-rank at least 17, then the centralizer of some involution of G is in standard form. Our theorem actually asserts that the corresponding standard component satisfies conditions similar to those which hold in the groups of Lie type of odd characteristic. Thus, in effect, our result reduces the further study of such simple groups to the following general problem: Determine all simple groups in which the centralizer of some involution is in standard form with standard component of Lie type of odd characteristic. This statement is simply a more precise formulation of the general question which we discussed at the beginning: characterize the simple groups in terms of the structure of the centralizers of their involutions. Indeed, apart from a few degenerate cases of low Lie rank, every group of Lie type of odd characteristic possesses an involution whose centralizer is in standard form. As indicated before, considerable progress has been made in this whole area and there is reasonable hope that the entire problem can be completely solved. If and when this is accomplished, our theorem could then be used as a basis for an inductive characterization of the groups of Lie type of odd characteristic. To complete such a characterization, it would be necessary, in addition, to determine all simple A-groups of normal 2-rank less than 13. Although some of our general arguments break down in such cases, there exist a number of special methods for handling the difficulties that arise. We note also that our result in the odd characteristic case gives further evidence that the sporadic simple groups are somehow more related to the groups of Lie type of even characteristic. We shall conclude now with a few remarks concerning the nature of the proof of the two stated theorems. Let S be a Sylow 2-subgroup of the group G satisfying the respective conditions. In each instance one proceeds by contradiction and the entire aim of the analysis is to demonstrate that the group is a proper subgroup of G. < C G (x) x ranging over the involutions of S > Once this is established, the theorem in question follows immediately from a theorem of Bender. Indeed, the preceding assertion implies that G contains what is called a strongly embedded subgroup and Bender has completely classified all such groups. In particular, he has shown that PSL(2, 2"), Sz(2"), and PSU(3 9 2 n ) are the only simple groups which possess a strongly embedded subgroup. Thus the proof of both theorems comes down to what we may call " piecing together " the centralizers of the involutions of S. This will explain why our analysis requires conditions primarily on the centralizers of involutions. Furthermore, the need for S to contain abelian subgroups of suitably high rank comes about from the fact that we must continually compare the centralizers of different involutions of S and some degree of freedom is required to carry this out effectively. The entire piecing together process is very general, most of it being almost formal in nature. In fact, I have come to think of the main steps in the argument as being

6 360 D. GORENSTEIN B3 essentially " functorial ". Indeed, the proof rests ultimately on what I have previously termed a signalizer functor and the so-called signalizer functor theorem. If A is an elementary abelian 2-subgroup of the group G, we say that 9 is an A-signalizer functor on G if for each involution a in A there is associated an ^-invariant subgroup 9(C G (a)) of 0(C G (a)) which satisfies the compatibility condition for any pair of involutions a, b of A. 9(C G (a)) n C G (b) ç= 9(C G (b)) The signalizer functor theorem asserts that if A has rank at least 3, then the subgroup is of odd order. < 9(C G (a)) a ranging over the involutions of A > Since, in practice, G will be non-solvable, this result implies that the given subgroup is a proper subgroup of G. David Goldschmidt has recently given an improved version and much simpler proof of the signalizer functor theorem than the original one that appears in the Journal of Algebra. We note finally that under the assumptions of our first theorem, it turns out that 9(C G (a)) = 0(C G (a)) defines an 4-signalizer functor on G. The aim of the proof is then to show that this 9 is, in fact, the trivial signalizer functor. From this, the desired conclusion follows easily. In summary, we have attempted to indicate that in simple groups whose proper subgroups have composition factors of known type and whose Sylow 2-subgroups are suitably large, general methods exist which enable one to determine, at least partially, the structure of the centralizers of their involutions. Rutgers University Department of Mathematics, New Brunswick, New Jersey (U. S. A.)