**John Thompson** studied at Yale University, receiving his B.A. in 1955. He went to the University of Chicago to undertake research and completed his doctorate in 1959. His doctoral thesis, entitled *A Proof that a Finite Group with a Fixed-Point-Free Automorphism of Prime Order is Nilpotent* was supervised by Mac Lane. In fact his doctoral thesis solved one of the conjectures of Frobenius which had remained unsolved for around 60 years. Thompson's thesis, as is clear from its title, proved Frobenius's conjecture that a finite group with an automorphism which does not fix any group element is necessarily nilpotent.

The solution of Frobenius's conjecture was not done by simply pushing the existing techniques further than others had done; rather it was achieved by introducing many highly original ideas which were to lead to many developments in group theory.

Thompson was an assistant at Harvard University in 1961-62, then, in 1962, he was appointed professor at the University of Chicago. In 1968 Thompson accepted a fellowship at University College, Cambridge in England. He was appointed Rouse Ball Professor of Pure Mathematics at Cambridge in 1970.

It is no coincidence that starting at the time of Thompson's thesis, group theory leapt into prominence as the mathematical topic which was attracting most attention and which was undergoing the most rapid development. The reason was that suddenly progress began to be made on one of the main problems of finite group theory, namely the classification of finite simple groups.

Every finite group can be viewed as built from a finite collection of finite simple groups. The finite simple groups are therefore the building blocks from which finite groups are built. To classify finite groups therefore reduces to two problems, namely the classification of finite simple groups and the solution of the extension problem, that is the problem of how to fit the building blocks together.

Early contributions were made by Galois, Jordan and Émile Mathieu. Claude Chevalley showed in 1955 that the Lie groups have finite analogues which are finite simple groups. M Suzuki in 1960 discovered new infinite families of finite simple groups. These were discovered by him independently of Chevalley's theory but then it was noticed that they were indeed twisted Chevalley groups. An automorphism had been missed in the original working out of Chevalley's theory which is why the Suzuki groups were only discovered some time after.

Thompson, working with Walter Feit, proved in 1963 that all nonabelian finite simple groups were of even order. They published this result in *Solvability of Groups of Odd Order* a 250 page paper which appeared in the *Pacific Journal of Mathematics* **13** (1963), 775-1029. Despite the importance of the paper several journals declined to publish it because of its length. The paper consists of one whole part of Volume 13 of the *Pacific Journal*. This result stunned the world of mathematics but it also led mathematicians to believe that a classification of finite simple groups might prove possible. Both Thompson and Feit received the Frank Nelson Cole Prize in 1965 when the thirteenth award was made to them for this their joint paper.

Another major early step by Thompson towards the classification of finite simple groups was his classification of those finite simple groups in which every soluble subgroup has a soluble normaliser.

Thompson was awarded a Fields Medal for his work at the International Congress of Mathematicians in Nice in 1970. Brauer, speaking of Thompson's work at the Congress, first spoke of the 'odd order paper':-

The first paper I have to mention is a joint paper by Walter Feit and John Thompson and, of course, Feit's part in it should not be overlooked. Here, the authors proved a famous conjecture, to the effect that all non-cyclic finite simple groups have even order. I am not sure who was the first to observe this. Fifty years ago[1920]this was already referred to as a very old conjecture. While it was usually mentioned in courses on algebra, it is only fair to say that nobody ever did anything about it, simply because nobody had any idea how to get started. It was not even clear that the whole problem made sense. Was the role of the prime2simply a little accident; did2play an entirely exceptional role, or were there properties of other prime divisors of the group order which bore at least some resemblance to those of2? It was only after the Feit-Thompson paper that one could be sure that the whole question was a reasonable one.

Brauer went on to speak of Thompson's subsequent work:-

Thompson's work which has now been honoured by the Fields medal is a sequel to this first paper. In it he determined the minimal simple finite groups, this is to say, the simple groups whose proper subgroups are solvable. Actually, a more general problem is solved. It suffices to assume that only certain subgroups, the so-called local subgroups, are solvable. These are the normalizers of the subgroups of prime power order ... These results are the first substantial results achieved concerning simple groups. A number of important corollaries show that one is now able to answer questions on finite groups which were completely out of reach before. I mention one: a finite group is solvable if and only if every subgroup generated by two elements is solvable.

The nonabelian finite simple groups fall into a small number of infinite series and 26 sporadic groups. During the 1970s Thompson contributed to the understanding of these groups. Brauer, in a personal comment at the end of [3] predicted this:-

On reaches a point in life where one wonders what one still expects of life, what one would still like to see happen. This applies to mathematics too. I have passed the point I have mentioned. I like to say that I would like to see the solution of the problem of the finite simple groups and the part I expect Thompson's work to play in it. Quite generally I would like to see to what further heights Thompson's future work will take him.

John Thompson's interests after 1970 became broader and over the 1970s he also made major contributions to coding theory. His work on coding theory was to lay the foundation for the solution of a long standing problem, namely the fact that there is no finite plane of order 10.

During the 1980s much of Thompson's work was on the problem of which finite groups could occur as Galois groups. Work in this area was started by Hilbert with his proof of the irreducibility theorem, and the authors of [4] state that:-

Thompson's work may well be the most important advance since Hilbert's time.

In 1989 Thompson was one of the five main speakers at the Groups St Andrews meeting. He gave a series on lectures on *Galois groups* at that meeting. The picture of Thompson shown here was taken in St Andrews during the conference.

Thompson has received many awards for his outstanding contributions to mathematics. In addition to the Cole Prize from the American Mathematical Society and the Fields Medal in 1970 described above, he was awarded the Senior Berwick Prize from the London Mathematical Society in 1982, the Sylvester Medal from the Royal Society in 1985 and he received the Wolf Prize and the Poincaré Prize in 1992. He was elected to the National Academy of Sciences in the United States in 1971 and the Royal Society of London in 1979. He was awarded the National Medal of Science in 2000.

Among the honorary degrees that Thompson has received are ones from Yale University (1980), the University of Chicago (1985), the University of Oxford (1987) and Ohio State University (2008).

In 2008 the Norwegian Academy of Science and Letters awarded the Abel Prize to John Griggs Thompson and Jacques Tits:-

... for their profound achievements in algebra and in particular for shaping modern group theory.

The Press Release gives the following summary of Thompson's contributions:-

Thompson revolutionised the theory of finite groups by proving extraordinarily deep theorems that laid the foundation for the complete classification of finite simple groups, one of the greatest achievements of twentieth century mathematics. Simple groups are atoms from which all finite groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an even number of elements. Later Thompson extended this result to establish a classification of an important kind of finite simple group called an N-group. At this point, the classification project came within reach and was carried to completion by others. Its almost incredible conclusion is that all finite simple groups belong to certain standard families, except for26sporadic groups. Thompson and his students played a major role in understanding the fascinating properties of these sporadic groups, including the largest, the so-called Monster. The achievements of John Thompson and of Jacques Tits are of extraordinary depth and influence. They complement each other and together form the backbone of modern group theory.

**Article by:** *J J O'Connor* and *E F Robertson*

**September 2009**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Thompson_John.html]