约翰·格里格斯·汤普森John G. Thompson

约翰·格里格斯·汤普森在耶鲁大学学习，1955年获得学士学位。他前往芝加哥大学进行研究，并于1959年完成博士学位。他的博士论文题为A Proof that a Finite Group with a Fixed-Point-Free Automorphism of Prime Order is Nilpotent，由桑德斯·麦克兰恩指导。事实上，他的博士论文解决了费迪南德·格奥尔格·弗罗贝尼乌斯的一个猜想，该猜想已经悬而未决约60年。汤普森的论文，从其标题就可以看出，证明了费迪南德·格奥尔格·弗罗贝尼乌斯的猜想：一个有限群，如果有一个不固定任何群元素的自同构，则必然是幂零的。

费迪南德·格奥尔格·弗罗贝尼乌斯猜想的解决并不是通过简单地将现有技术推进得比别人更远而完成的；而是通过引入许多高度原创的想法来实现的，这些想法导致了群论中的许多发展。

汤普森于1961-62年在哈佛大学担任助理，随后于1962年被任命为芝加哥大学教授。1968年，汤普森接受了英格兰剑桥大学学院的研究员职位。他于1970年被任命为剑桥大学Rouse Ball纯数学讲席教授。

从汤普森的学位论文开始，group theory作为最受关注、发展最迅速的数学主题而崭露头角，这并非巧合。原因在于，有限群论的主要问题之一，即有限单群的分类，突然开始取得进展。

每个有限群都可以看作是由有限个有限单群构建而成的。因此，有限单群是构建有限群的基本构件。因此，有限群的分类归结为两个问题，即有限单群的分类和扩张问题的解决，也就是如何将这些基本构件组合在一起的问题。

早期的贡献由埃瓦里斯特·伽罗瓦、卡米耶·若尔当和以米里迂·拉·马丢做出。Claude 谢瓦莱在1955年证明了李群具有有限类似物，即有限单群。M Suzuki在1960年发现了新的无限族有限单群。这些是他独立于谢瓦莱的理论发现的，但随后人们注意到它们确实是扭谢瓦莱群。在最初推导谢瓦莱的理论时遗漏了一个自同构，这就是为什么Suzuki群直到一段时间之后才被发现。

汤普森与瓦尔特·法伊特合作，于1963年证明了所有非阿贝尔有限单群都具有偶数阶。他们将这一结果发表在Solvability of Groups of Odd Order中，这是一篇250页的论文，发表于Pacific Journal of Mathematics 13 (1963), 775-1029。尽管这篇论文很重要，但几家期刊因其篇幅而拒绝发表。这篇论文构成了Pacific Journal第13卷的整整一部分。这一结果震惊了数学界，但也使数学家们相信有限单群的分类可能是可行的。汤普森和瓦尔特·法伊特于1965年获得了法兰克·尼尔森·寇尔奖，当时第十三届奖项授予他们，以表彰他们这篇合著论文。

汤普森在有限单群分类方面取得的另一个早期重大进展是他对那些每个soluble subgroup都具有可解正规化子的有限单群进行了分类。

汤普森因他在1970年尼斯国际数学家大会上的工作而获得了菲尔兹奖。理查德·布饶尔在谈到汤普森在大会上的工作时，首先提到了“奇数阶论文”：-

我必须提到的第一篇论文是瓦尔特·法伊特和汤普森的合著论文，当然，瓦尔特·法伊特在其中的贡献不应被忽视。在这里，作者们证明了一个著名的猜想，即所有非循环有限单群都有偶数阶。我不确定是谁首先观察到这一点。五十年前[1920年]，这已经被认为是一个非常古老的猜想。虽然它通常在代数课程中被提及，但公平地说，从来没有人对此做过任何事情，仅仅是因为没有人知道如何开始。甚至不清楚整个问题是否有意义。素数2的作用仅仅是一个小意外吗；2是否扮演了一个完全特殊的角色，还是群阶的其他素因子的性质至少与2有一些相似之处？只有在瓦尔特·法伊特-汤普森的论文之后，人们才能确信整个问题是一个合理的问题。

理查德·布饶尔接着谈到了汤普森随后的工作：-

汤普森的工作现在获得了约翰·查尔斯·菲尔兹奖章的荣誉，这是第一篇论文的续篇。在其中，他确定了极小有限单群，也就是说，其真子群可解的单群。实际上，解决了一个更一般的问题。只需假设只有某些子群，即所谓的局部子群，是可解的。这些是素数幂阶子群的正规化子……这些结果是关于单群取得的第一个实质性结果。许多重要的推论表明，现在能够回答以前完全无法触及的有限群问题。我提到一个：一个有限群是可解的当且仅当由两个元素生成的每个子群都是可解的。

非阿贝尔有限单群分为少数几个无限系列和26个散在群。在20世纪70年代，汤普森对这些群的理解做出了贡献。理查德·布饶尔在[3]末尾的个人评论中预测了这一点：-

人在一生中会到达一个阶段，那时他会想自己还对生活期待什么，还希望看到什么发生。这也适用于数学。我已经过了我提到的那个阶段。我喜欢说，我希望看到有限单群问题的解决，以及我期待汤普森的工作在其中所起的作用。总的来说，我希望看到汤普森未来的工作将把他带到什么样的更高高度。

1970年后，汤普森的兴趣变得更加广泛，在20世纪70年代，他还在编码理论方面做出了重大贡献。他在编码理论方面的工作为长期存在的问题的解决奠定了基础，即不存在阶为10的有限平面这一事实。

在20世纪80年代，汤普森的大部分工作都集中在哪些有限群可以作为Galois groups出现的问题上。这一领域的工作由大卫·希尔伯特开始，他证明了不可约性定理，而[4]的作者们指出：-

汤普森的工作很可能是自大卫·希尔伯特时代以来最重要的进展。

1989年，汤普森是Groups St Andrews会议的五位主要演讲者之一。他在那次会议上就Galois groups做了一系列讲座。这里展示的汤普森的照片是在会议期间于圣安德鲁斯拍摄的。

汤普森因其对数学的杰出贡献而获得了许多奖项。除了上述来自美国数学会的法兰克·尼尔森·寇尔奖和1970年的菲尔兹奖章外，他还在1982年获得了来自伦敦数学会的高级伯威克奖，1985年获得了来自皇家学会的詹姆斯·约瑟夫·西尔维斯特奖章，并在1992年获得了沃尔夫奖和儒勒·昂利·庞加莱奖。他于1971年当选为美国的国家科学院，1979年当选为伦敦皇家学会。他于2000年获得了国家科学奖章。

汤普森获得的荣誉学位包括耶鲁大学（1980年）、芝加哥大学（1985年）、牛津大学（1987年）和俄亥俄州立大学（2008年）的荣誉学位。

2008年，Norwegian Academy of Science and Letters将尼尔斯·阿贝尔奖授予弗瑞兹·约翰 Griggs 汤普森和雅克·蒂茨：-

……以表彰他们在代数方面的深刻成就，特别是为塑造现代群论所做的贡献。

新闻稿对汤普森的贡献作了如下总结：-

汤普森通过证明极其深刻的定理，彻底改变了有限群论，为有限单群的完全分类奠定了基础，这是20世纪数学最伟大的成就之一。单群是构成所有有限群的基本单元。在一项重大突破中，瓦尔特·法伊特和汤普森证明了每个非初等单群都有偶数个元素。后来，汤普森扩展了这一结果，建立了一类重要的有限单群——称为$N$群——的分类。至此，分类项目已触手可及，并由其他人完成。其几乎令人难以置信的结论是，所有有限单群都属于某些标准族，除了26个散在群。汤普森和他的学生在理解这些散在群的迷人性质方面发挥了重要作用，包括最大的所谓怪物群。汤普森和雅克·蒂茨的成就是非凡的深度和影响力。它们相互补充，共同构成了现代群论的支柱。

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 prime 2 simply a little accident; did 2 play an entirely exceptional role, or were there properties of other prime divisors of the group order which bore at least some resemblance to those of 2? 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 for 26 sporadic 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.