mem-suzuki.qxp 3/8/99 10:09 AM Page 543 Michio Suzuki (1926–1998) Michael Aschbacher, Helmut Bender, Walter Feit, and Ronald Solomon Editor’s Note. Michio Suzuki, an early leader in the effort to classify finite simple groups, died May 31, 1998, in Tokyo at the age of seventy-one. Born October 2, 1926, in Japan, he obtained his Ph.D. from the University of Tokyo in 1952, with Shoukichi Iyanaga as official advisor. Suzuki’s teachers included also Yasuo Akizuki and Kenkichi Iwasawa. Suzuki assumed a faculty position at the University of Illinois, Urbana-Champaign, beginning the next year. In 1956–57 he took a leave of absence to work at Harvard University as research associate with Richard Brauer, with support from the National Science Foundation. He was a professor in the Center for Advanced Study at the University of Illinois from 1968 until his death. Suzuki held a postdoctoral fellowship in 1952–53 and a Guggenheim Fellowship in 1962–63, received the Acad- emy Prize from the Japan Academy in 1974 for his work in group theory, and was awarded an honorary doctoral degree from the University of Kiel, Germany, in 1991. He had visiting appointments at the University of Chicago (1960–61); the Institute for Advanced Study in Princeton (1962–63, 1968–69, and spring 1981); the University of Tokyo (spring 1971); the Universities of Hokkaido, Osaka, and Tokyo (1981 and 1985); and the University of Padua, Italy (1994). Walter Feit order indicates why L(G × G) should always be much richer than L(G).) Michio Suzuki was one of the group of brilliant During the summer of 1952 he came to Ann young Japanese mathematicians who entered col- Arbor, attracted by the presence of Richard Brauer, lege after World War II. He received his Ph.D. in who was on the faculty there. Brauer was one of 1952 from the University of Tokyo in absentia. the very few senior mathematicians in the USA Prior to that he came to the University of Illinois who worked on questions concerning the structure in 1952 as a research fellow. He joined the faculty of finite simple groups. He and Brauer ran a sem- of the University in 1953, a position he held until inar that summer, which John Walter and I and oth- his sudden death. ers attended. I met him in that seminar while I was His thesis was in the theory of finite groups, and a graduate student at the University of Michigan. this subject was to occupy him for his whole ca- The theory of finite groups became a subject of reer. His early work included a study of the lattice intensive research during the next few years. One of all subgroups L(G) of a group G. He proved that reason was John Thompson’s thesis, which intro- if G is a noncyclic simple finite group and H is a duced new methods and ideas to the subject; an- finite group with L(G × G)=L(H × H), then G is other was the progress in character theory sparked isomorphic to H. At the time it was not known by Brauer and Suzuki. whether L(G) determines G up to isomorphism. It is necessary here to make some definitions. By way of background, an important theorem due However, by using the classification of the finite to Frobenius says that if H is a finite transitive per- simple groups, it is possible to prove the more nat- mutation group such that the subgroup fixing a let- ural result that if G and H are noncyclic finite sim- ter is nontrivial and no nonidentity element fixes ple groups with L(G)=L(H), then G is isomorphic two or more letters, then H contains a proper non- to H. (Consideration of a cyclic group G of prime trivial normal subgroup M such that every non- Walter Feit is professor of mathematics at Yale University. identity element x in M has centralizer CH (x) con- His e-mail address is [email protected]. tained in M. All known proofs of this theorem use Photographs courtesy of Koichiro Harada and Naoko character theory. We use this theorem to make a Suzuki. definition: A finite group H is a Frobenius group MAY 1999 NOTICES OF THE AMS 543 mem-suzuki.qxp 3/8/99 10:09 AM Page 544 with Frobenius kernel M if orem defines a one-to-one correspondence be- M is a proper nontrivial tween the set of irreducible characters of H that normal subgroup of H and do not have M in their kernel and a set of irre- if every nonidentity element ducible characters of G, the exceptional characters x in M has centralizer of G. The construction of the exceptional charac- CH (x) contained in M. ters makes it possible to derive information con- A subset S of a group G cerning their values on certain elements of G. In is a trivial intersection set in this way it is possible to construct a fragment of G (briefly, a T.I. set in G) if the character table of G. In some circumstances −1 ∩ ⊆{ } x Sx S 1 for every this fragment of the character table is enough to −1 6 x in G with x Sx = S. yield significant information about the group G. Suppose that G contains This formulation has one fundamental advan- a subgroup H that is a tage over Brauer’s original observation, in that it Frobenius group with an is not necessary to begin with an irreducible char- abelian Frobenius kernel M. acter of G. Rather, it implies the existence of ir- Assume that H is the nor- reducible characters of G with certain properties. malizer of M and M is a T.I. That is precisely the essence of this approach. Michio Suzuki, December 1994, set in G. Let |M| = m, and Manila, Phillipines. |H| = me. Then e | (m − 1) In 1957 Suzuki [Su3] showed that a group of odd and H has exactly order in which the centralizer of every nonidentity (m − 1)/e irreducible characters that do not con- element is abelian is solvable. If it is assumed that tain M in their kernel. Furthermore, they all have G is a counterexample of minimum order, then it degree e. is not difficult to show that G is simple and every Brauer used his characterization of characters maximal subgroup of G is a Frobenius group H to observe the following. and the hypotheses of the theorem are satisfied | | Let χ1 be an irreducible character of G, and let for H. Hypothesis (ii) follows from the fact that G 1 and 2 be irreducible characters of H which do is odd. Thus the theorem shows the existence of not have M in their kernel such that 1 and χ1 exceptional characters corresponding to H. Suzuki agree on M −{1}. Let χ2 be derived from χ1 by showed that every irreducible nonprincipal char- replacing 1 by 2 on the conjugates of M −{1}. acter of G is exceptional for some maximal sub- Then χ2 is an irreducible character of G. group of G. He was then able to reach a contra- He told me that he wrote to Suzuki about this diction to the assumed existence of G by using and immediately received by return mail Suzuki’s various properties of exceptional characters. alternative proof. (Presumably Suzuki had inde- This paper is a gem! At the time its importance pendently been thinking along these lines.) was not fully grasped, either by him or by others, In the above notation suppose that e =6 m − 1. as it seemed to be simply an elegant exercise in Then there are at least two faithful irreducible character theory. However, the result and the meth- characters 1, 2 of H that do not have M in their ods used had a profound impact on much suc- kernel. It is easily seen that ceeding work, and in particular it is an essential ingredient of the work by John Thompson and me k G − Gk2 k − k2 1 2 G = 1 2 H =2. on groups of odd order. In fact it was the first sig- nificant result on groups of odd order since Burn- This argument can be used to prove the following side’s work. Much of his later work had a more im- result due to Brauer and Suzuki. mediate impact and so overshadowed this paper, Theorem. Let H,M,m,e be as above. Assume that which is perhaps even today still not fully appre- ciated. (i) M is a T.I. set. By Thompson’s thesis M is always nilpotent if (ii) m − 1 =6 e. H is a Frobenius group as above. Soon thereafter Marshall Hall, John Thompson, and I were able to (iii) M is abelian. show that if G is a group of odd order in which Let {i} be the set of irreducible characters of H the centralizer of every nonidentity element is that do not have M in their kernel. Then there exist nilpotent, then G is solvable. The proof consisted a sign ε = 1 and a set {χi} of irreducible char- of two parts: The first part was an application of acters of G such that Thompson’s methods to reduce the problem to the case that every proper maximal subgroup of G is G − G − i j = ε(χi χj ) a Frobenius group whose kernel is a T.I. set in G. Unlike the abelian case, this was not easy to prove.