**Richard Brauer**'s father was Max Brauer who was a well-off businessman in the wholesale leather trade. Max Brauer's wife was Lilly Caroline and Richard was the youngest of their three children. He had an older brother Alfred Brauer, who also became a famous mathematician and has a biography in this archive. Alfred Brauer was seven years older than Richard and of an age between the two brothers was Richard's sister Alice.

Richard entered the Kaiser-Friedrich-Schule in Charlottenburg in 1907. Charlottenburg was a district of Berlin which was not incorporated into the city until 1920. Richard studied at this school until 1918 and it was during his school years that he developed his love of mathematics and science. However, this was not due to the teaching at the school, but it came about through the influence of his brother Alfred. In [4] Richard writes about his school teachers who he describes as not being very competent. There was one exception however, and it was fortunate that this good teacher was a mathematician who had a doctorate awarded for research done under Frobenius's supervision.

Of course Richard's last four years at school were the years of World War I, but, unlike his brother, he was young enough to avoid being drafted into the army. When he graduated from the Kaiser-Friedrich-Schule in September 1918 the war was still in progress, and Brauer was drafted to undertake civilian war service in Berlin. Only two months later, in November 1918, the war ended, Brauer was released from war service and he resumed his education. Despite the love for mathematics which he had gained from his brother, Brauer decided to follow his boyhood dreams of becoming an inventor. He entered the Technische Hochschule of Charlottenburg in February 1919 where he studied for a term before, having realised that his talents were in theory rather than practice, he transferred to the University of Berlin.

At the University of Berlin Brauer was taught by a number of really outstanding mathematicians including Bieberbach, Carathéodory, Einstein, Knopp, von Mises, Planck, Schmidt, Schur and Szegö. In [4] Brauer describes some of the lectures he attended; talking of Schmidt's lectures he writes:-

It is not easy to describe their fascination. When Schmidt stood in front of a blackboard, he never used notes, and was hardly ever well prepared. He gave the impression of developing the theory right there and then.

It was the custom that German students at this time spent periods in several different universities during their degree course. Brauer was no exception to this, although he made only one visit during his studies, that being for a term to the University of Freiburg. Back in Berlin he attended seminars by Bieberbach, Schmidt and Schur. He was increasingly attracted towards the algebra which Schur was presenting in his seminar (which was attended in the same year by Alfred Brauer). Schur, unlike Schmidt, [4]:-

... was very well prepared for his classes, and he lectured very fast. If one did not pay the utmost attention to his words, one was quickly lost. There was hardly any time to take notes in class; one had to write them up at home. ... He conducted weekly problem hours, and almost every time he proposed a difficult problem. Some of the problems had already been used by his teacher Frobenius, and others originated with Schur. Occasionally he mentioned a problem he could not solve himself.

In fact it was one of these open problems which Richard working with his brother Alfred solved in 1921 and this was eventually to be included in Brauer's first publication. Schur suggested the problem that Brauer worked on for his doctorate and the degree was awarded (with distinction) in March 1926. His dissertation took an algebraic approach to calculating the characters of the irreducible representations of the real orthogonal group. Before the award of his doctorate, however, Brauer had married Ilse Karger in September 1925. They had been a fellow students in one of Schur's courses on number theory. Before his marriage Brauer was appointed as Knopp's assistant at the University of Königsberg and he took up this post in the autumn of 1925.

Shortly after Brauer arrived in Königsberg, Knopp left to take up an appointment at Tübingen. The mathematics department at Königsberg was small, with two professors Szego and Reidemeister, and with Rogosinski and Kaluza holding junior positions like Brauer. It was in Königsberg that Brauer's two sons, George Ulrich Brauer and Fred Günter Brauer were born. Brauer taught at Königsberg until 1933 and during this period he produced results of fundamental importance. Green writes in [6] (or see [1]):-

This was the time when Brauer made his fundamental contribution to the algebraic theory of simple algebras. ... Brauer developed ... a theory of central division algebras over a given perfect field, and showed that the isomorphism classes of these algebras can be used to form a commutative group whose properties gave great insight into the structure of simple algebras. This group became known(to the author's embarrassment)as the "Brauer group"...

Political events forced Brauer's family to move. He wrote (see [4]):-

I lost my position in Königsberg in the spring of1933after Hitler became Reichskanzler of Germany.

Brauer was from a Jewish family so was dismissed from his post under the Nazi legislation which removed all Jewish university teachers from their posts. This was a desperate time for Brauer who realised that he had to find a post outside Germany. Fortunately action was taken in several countries to find posts abroad for German academics forced from their positions and a one year appointment was arranged for Brauer in Lexington, Kentucky for the academic year 1933-34. In November 1933 Brauer arrived to take up his appointment at the University of Kentucky, his wife and two sons following three months later. We should record that Alfred Brauer left Germany in 1939, but Brauer's sister Alice stayed behind and was murdered in a concentration camp by the Nazis.

Following his year in Lexington, Brauer was appointed as Weyl's assistant at the Institute for Advanced Study in Princeton. He wrote in [4] about this appointment with Weyl:-

I had hoped since the days of my PhD thesis to get in contact with him some day; this dream was now fulfilled.

Collaboration between Brauer and Weyl on several projects followed, in particular a famous joint paper on spinors published in 1935 in the *American Journal of Mathematics*. This work was to provide a background for the work of Paul Dirac in his exposition of the theory of the spinning electron within the framework of quantum mechanics.

A permanent post followed the two temporary posts when Brauer accepted an assistant professorship at the University of Toronto in Canada in the autumn of 1935. It was largely as a result of Emmy Noether's recommendation, which she made while visiting Toronto, which led to his appointment. This was a time when Brauer developed some of his most impressive theories, carrying the work of Frobenius into a whole new setting, in particular the work on group characters Frobenius published in 1896. Brauer carried Frobenius's theory of ordinary characters (where the characteristic of the field does not divide the order of the group) to the case of modular characters (where the characteristic does divide the group order). He also studied applications to number theory.

C J Nesbitt was Brauer's first doctoral student in Toronto and he described their relationship as doctoral student and supervisor (see for example [6]):-

Curiously, as thesis advisor, he did not suggest much preparatory reading or literature search. Instead we spent many hours exploring examples of the representation theory ideas that were evolving in his mind.

It was in joint work with Nesbitt, published in 1937, that Brauer introduced the theory of blocks. This he used to obtain results on finite groups, particularly finite simple groups, and the theory of blocks would play a big part in much of Brauer's later work.

Alperin also spoke of Brauer's thirteen years in Toronto (see [3]):-

The years he spent at Toronto were his most productive years. He achieved five or six great results during that time, any one of which would have established a person as a first-rank mathematician for the rest of their life. ... those years had their high points, but also contained fallow periods, when there was the day-to-day grind of raising a family in modest circumstances.

Brauer spent 1941 at the University of Wisconsin having been awarded a Guggenheim Memorial Fellowship. He was the Colloquium lecturer at the American Mathematical Society Summer Meeting in Madison, Wisconsin in 1948. Later that year he moved from Toronto back to the United States, accepting a post at the University of Michigan in Ann Arbor. In 1949 Brauer was awarded the Cole Prize from the American Mathematical Society for his paper *On Artin's L-series with general group characters* which he published in the *Annals of Mathematics* in 1947. In 1951 Harvard University offered him a chair and, in 1952, he took up the position in Harvard which he was to hold until he retired in 1971. In the year of his retirement he was honoured with the award of the National Medal for Scientific Merit.

We have mentioned a number of topics which Brauer worked on in the course of this biography. However we have not yet mentioned the work which in many ways was his most famous and this he began around the time he took up the chair at Harvard. He began to formulate a method to classify all finite simple groups and his first step on this road was a group-theoretical characterisation of the simple groups PSL(2,q) in 1951 (although for a complicated number of reasons explained in [6] and [1] this did not appear in print until 1958). This work was done jointly with his doctoral student K A Fowler, and in 1955 they published a major paper which was to set mathematicians on the road to the classification. The paper was *On groups of even order* and it provided the key to the major breakthrough by Walter Feit and John Thompson when they proved that every finite simple group has even order.

Brauer was to spend the rest of his life working on the problem of classifying the finite simple groups. He died before the classification was complete but his work provided the framework of the classification which was completed only a few years later. (See the biography of Gorenstein for further details on the programme to classify finite simple groups.) Most important was Brauer's vital step in setting the direction for the whole classification programme in the paper *On groups of even order* where it is shown that there are only finitely many finite simple groups containing an involution whose centraliser is a given finite group. Brauer had announced these results and his programme for classifying finite simple groups at the International Congress of Mathematicians in Amsterdam in 1954.

Green in [6] points out that when Brauer went to Harvard he was 51 years old, yet almost half his total of 147 publications were published after this date. He certainly did not sit quietly working away in Harvard. He spent extended periods visiting friends and colleagues in universities around the world, for example Frankfurt and Göttingen in Germany, Nagoya in Japan, and Newcastle and Warwick in England.

Despite his remarkable contributions to research, Brauer found time to act as an editor for a number of journals. He was an editor of the *Transactions of the Canadian Mathematical Congress* from 1943 to 1949, the *American Journal of Mathematics* from 1944 to 1950, the *Canadian Journal of Mathematics* from 1949 to 1959, the *Duke Mathematical Journal* from 1951 to 1956 and again from 1963 to 1969, the *Annals of Mathematics* from 1953 to 1960, the *Proceedings of the Canadian Mathematical Congress* from 1954 to 1957, and the *Journal of Algebra* from 1964 to 1970. A quick glance will show that in 1955 he held editorships of four learned journals.

We have mentioned above a number of honours which Brauer received. We should also mention the learned societies which honoured him with membership: the Royal Society of Canada (1945), the American Academy of Arts and Sciences (1954), the National Academy of Sciences (1955), the London Mathematical Society (1963), the Akademie der Wissenschaften Göttingen (1964), and the American Philosophical Society (1974). He was also elected President of the Canadian Mathematical Congress (1957-58) and the American Mathematical Society (1959-60).

In [1] and [6] Green describes Brauer's character (no pun intended!):-

All who knew him best were impressed by his capacity for wise and independent judgement, his stable temperament and his patience and determination in overcoming obstacles. He was the most unpretentious and modest of men, and remarkably free of self-importance....

Brauer's interest in people was natural and unforced, and he treated students and colleagues alike with the same warm friendliness. In mathematical conversations, which he enjoyed, he was usually the listener. If his advice was sought, he took this as a serious responsibility, and would work hard to reach a wise and objective decision.

Richard Brauer occupied an honoured position in the mathematical community, in which the respect due to a great mathematician was only one part. He was honoured as much by those who knew him for his deep humanity, understanding and humility; these were the attributes of a great man.

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

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