**Marshall Hall Jr.**'s parents were Marshall Hall and Inez Bethune. Marshall Hall Sr. was educated in private schools in Kentucky and was first employed by the Chesapeake & Ohio Railroad Company. He took up employment with the W L Green Commission Company, an old and prominent grain company of St Louis, in 1894 and he married Inez Bethune on 27 April 1897 in St Louis. They had four children: Sydnor. Bethune, Marshall, Jr. (the subject of this biography), and Inez. Marshall Hall Sr. worked his way up to become president of the W L Green Commission Company in 1903. In 1915 the company was renamed the Marshall Hall Grain Company, with Marshall Hall Sr. as its president. Marshall Hall Jr. showed talent for mathematics at a young age for, at age eleven, he constructed a seven-place table of logarithms for the positive integers up to 1000. A year later he felt that mathematics was his subject when [6]:-

... my history teacher dismissed my ideas as erroneous while my mathematics teacher listened and even agreed.

His high school studies were at the St Louis Country Day School. This school, founded in 1917, had only been open for a few years when he began studying there. The school was outside the city, in a rural setting, reached by electric streetcar. Marshall was taught mathematics by a good teacher although he was not a mathematician, being employed as a German teacher. While he was at this school Marshall's father died and, although the family had been very well off, the death of his father meant not only sadness at the loss, but also that the family were now in a poorer financial position.

He entered Yale as an undergraduate and there he took many advanced courses in topics such as calculus of variations and algebraic numbers. He did not study only science subjects at Yale, however, also taking classics courses. He graduated with a B.A. in 1932 having won a prize in ever one of his years at university. In his final year at Yale he applied to the Charles and Julia Henry Fund for a Henry Fellowship to enable him to undertake graduate studies abroad. This Fellowship was to fund one year of study at Oxford or Cambridge in England. He won the Fellowship and spent the year 1932-33 working with G H Hardy at Trinity College, Cambridge.

At Cambridge he was taught by several mathematicians who were to have an important influence on him such as Philip Hall, Harold Davenport and G H Hardy. He wrote his first paper *Quadratic residues in factorization* which he submitted to the *Bulletin* of the American Mathematical Society in March 1932. While he was at Cambridge, G H Hardy helped him edit the paper which was published in 1933. Hall writes in the Introduction:-

The purpose of this paper is to establish a certain theorem which is useful in the factorization of large numbers.

Hall would have liked to continue to undertake graduate work at Cambridge but since the fellowship was only for one year and his family could not support him financially, he applied for actuarial positions back in the United States. He was offered a position with an insurance company in St Louis which he accepted and began to take actuarial examinations so that he might qualify as an actuary. In the spring of 1934 he received two offers on consecutive days. One was for a position he had applied for at the Metropolitan Life Insurance Company while the other was the offer of a fellowship at Yale for graduate work, for which he had not applied. He accepted the offer of a fellowship at Yale and he was awarded his doctorate in 1936 for his thesis *An Isomorphism Between Linear Recurring Sequences and Algebraic Rings*. His official thesis advisor was Øystein Ore but he received a great deal of help and advice from Howard Theodore Engstrom (1902-1962).

After the award of his Ph.D., Hall spent the year 1936-37 at the Institute for Advanced Study at Princeton. he wrote in [6]:-

Fortunately G H Hardy also spent the year1936-37at the Institute and I helped edit his 'Lectures on Ramanujan'. This was very inspiring and among other things got me into analytic number theory. I got to know John von Neumann very well and attended his lectures on regular algebras and their associated geometric systems. He did not take well to my suggestion of calling them "pointless algebras." I made the acquaintance of Hermann Weyl, J H M Wedderburn, Marston Morse, Oswald Veblen and other luminaries. We shared space in Fine Hall with the Mathematics Department. I lived in the Graduate College within easy walking distance of everything that mattered.

While at the Institute for Advanced Study, Hall received offers of an Instructorship in Mathematics from Columbia University and a similar offer from Yale University. He accepted Yale and began working there in the autumn of 1937. He was made a fellow of Silliman College and lived in rooms in the College. Up to this time he had not been involved in research in group theory but it was at this stage in his career that his interest in this topic began, prompted by Øystein Ore. He also became interested in projective geometry and began a major study of projective planes. He presented his work to the American Mathematical Society in September 1940.

Another fellow of Silliman College was Judge Charles Edward Clark (1889-1963) who had been Dean of the Yale Law School from 1929 to 1939. He had married Dorothy E Gregory and Hall met their daughter Sally at a Silliman College picnic. They were married in June 1942. However, the Japanese attack on Pearl Harbor in December meant that the United States entered World War II and, as a consequence, Hall's life over the next few years was very different. With the United States now at war, Hall joined Naval Intelligence and was involved, as were many other mathematicians, deciphering Japanese coded messages and deciphering German Enigma codes. This work was little known about at the time but it has since emerged how significant this work proved to be. Hall's work in this area remained covered by the Official Secrets Act and when I [EFR] was in a group talking to him in 1981 about such things, he made it clear how seriously he continued to take his signing of the Act. Hall spent six months at Bletchley Park, the centre for British Intelligence, during 1944. At Bletchley [6]:-

... there was a galaxy of mathematical talent including Hugh Alexander the chess champion and Henry Whitehead the eminent topologist and Waynflete Professor at Magdalen College, Oxford.

Hall's 48-page paper *Projective planes* was published in the *Transactions* of the American Mathematical Society in 1943. In it he states:-

The author has made an attempt to investigate projective planes as systematically as possible. A considerable portion of these investigations is not included here either because of the incomplete and unsatisfactory nature of the results or because of lack of generality. War duties have forced the postponement of the completion of this work.

After World War II, Hall returned to Yale where he taught until 1946. However he discovered that the atmosphere at Yale was rather unpleasant and his association with Øystein Ore was making his position almost untenable [6]:-

... there was a feud between Øystein Ore and Einar Hille, who had married Ore's sister. Ore was the Sterling Professor and nothing could be done to him. But his enemies took it out on me, saying that in no circumstances could I be promoted and given tenure. One of the enemies, Nelson Dunford, made the amazing statement that my 'Projective Planes' paper was so good that he doubted that I could write another good paper.

Hall, therefore, accepted an appointment as an associate professor at Ohio State University and began teaching there in the autumn of 1946. Saunders Mac Lane had recommended Hall to Tibor Radó, the chairmen of the Ohio State Mathematics Department, who treated him very well. He was promoted to full professor at Ohio in 1948 and remained there until 1959. While on the faculty at Ohio he was awarded a Guggenheim Fellowship which enabled him to spend the year 1956 at Trinity College, Cambridge. There he and his family (by this time Hall had two young sons) lived in Louis Mordell's home. Hall is best known as a group theorist, perhaps because of his famous book *Theory of Groups* (1959) from which several generations of group theorists have learnt the subject. He was writing this book during his year in Cambridge and Philip Hall read his manuscript and made many helpful suggestions.

You can read short extracts from several reviews of this important book at THIS LINK

In 1959 he accepted a post at California Institute of Technology at Pasadena. He hosted a 'Conference on Group Theory' there in 1960. In the following year, he was involved in a joint project with NASA's Jet Propulsion Laboratory to construct a special Hadamard Matrix [4]:-

In1961, mathematicians from NASA's Jet Propulsion Laboratory and Caltech worked together to construct a Hadamard Matrix containing92rows and columns, with combinations of positive and negative signs. In a Hadamard Matrix, if you placed all the potential rows or columns next to each other, half of the adjacent cells would be the same sign, and half would be the opposite sign. This mathematical problem had been studied since about1893, but the solution to the92by92matrix was unproven until1961because it required extensive computation.

Hall spent time at the University of Warwick Group Theory year in August 1967 and, while in England, visited Philip Hall in Cambridge. It was at this time that he was able to confirm the existence of a simple group of order 604,800 which had been predicted by Zvoninir Janko. This group is now known as the Hall-Janko group. He published his findings in the paper *A search for simple groups of order less than one million* presented to the conference 'Computational Problems in Abstract Algebra' held in Oxford 29 August to 2 September 1967. He writes:-

Very recently Z Janko announced that a simple group with certain properties would have order604,800and have a specific character table. The construction of a simple group of order604,800is given for the first time in this paper. ... The construction of the simple group of order604,800was carried out in August1967at the University of Warwick and at Cambridge University. Mr Peter Swinnerton-Dyer was extremely helpful in writing on short notice a program for the Titan computer at Cambridge which finally confirmed the correctness of the construction.

It is worth noting that Janko's 'specific character table' for the group was shown to be in error by Walter Feit. The uniqueness of the simple group of this order was proved in Hall's paper, written jointly with David Wales, *The simple group of order 604,800* (1968). We should go back a little at this point and look at some of Hall's other contributions to group theory. Perhaps his best known result in group theory is his solution of the Burnside problem for groups of exponent 6. He showed that a finitely generated group in which the order of every element divides 6 must be finite. He outlined his proof in *Solution of the Burnside problem of exponent 6* (1957) and gave full details in the 22-page paper *Solution of the Burnside problem of exponent six* (1958). Around this time he also was involved in advising John Thompson [6]:-

John Thompson, then a graduate student at the University of Chicago, persuaded Saunders Mac Lane to invite me to talk on this subject[his solution of the Burnside problem for exponent6]. From then on John Thompson came to Columbus to work with me on a Ph.D. topic. I gave him the problem to prove that a group with an automorphism of prime order p fixing only the identity was necessarily nilpotent. This he did in fairly short order. It was an all or nothing assignment and never again have I given such an assignment. John's genius was soon recognised...

In collaboration with James Senior, Hall published *The Groups of Order *2^{n} (*n* ≤ 6) in 1964. Olga Taussky-Todd explains the background to this work [12]:-

The work was originally started separately by Senior(a chemist)and P Hall. Their later collaboration was interrupted by the war. When P Hall found himself unable to return to the plan, M Hall, Jr. used his wide experience in group theory to fill his place and to explain the theoretical background of the construction.

You can read short extracts from several reviews of this important book at THIS LINK

As well as the results on finite projective planes, Hall did other work of fundamental importance in the area of combinatorics, in particular on block designs. He wrote another classic text *Combinatorial Theory* in 1967. Giuseppe Pellegrino writes in a review that the book is:-

... a reference point for those interested in combinatorics, both for the selection of the topics - fruit of a deep knowledge of the field - and for the clearness of exposition that makes the reading agreeable.

You can read short extracts from several reviews of this important book at THIS LINK

This, and other contributions by Hall are mentioned by Hans Zassenhaus in [14]:-

The group theoretic collection process of Phillip Hall(no relation)inspired Marshall Hall to the construction of a basis for free Lie rings and higher commutators in free groups ... with great potential for further research ... The book on 'Combinatorial Theory'(1967and1986)summarizes his research achievements in combinatorics, in particular his deep results on combinatorial designs, and provides a new chart for an ancient branch of mathematics.

In 1973 Hall was named IBM Professor at CalTech but, in 1977, he arranged for his appointment at CalTech to be half-time so that he could spend time in other institutions. He was a Visiting Fellow at Merton College, Oxford in 1977, and at Technion, Haifa in 1980. He retired from CalTech in 1981 and, in the following year, accepted a post as Visiting Professor at Emory University in Atlanta. He was also a Visiting Professor at the University at Santa Barbara in 1984-85. Although up to this time he had kept his home in Pasadena, he moved to Atlanta in 1985. By this time he was on his own, having been divorced in 1981.

Among the honours Hall received were two Guggenheim Fellowships and membership of the American Academy of Arts and Sciences. He also received honorary degrees from Emery and from Ohio. The Marshall Hall Conference was held at the University of Vermont, Burlington, Vermont, September 13-18, 1990. The Preface to the Proceedings explains how it honours Marshall Hall, Jr.:-

From September13through September18, 1990, the 'Marshall Hall Conference on coding theory, design theory and group theory' was held on the campus of the University of Vermont. The multidisciplinary nature of this conference offered the190participants a unique opportunity to interact and discuss ideas with other researchers in these various areas. To our knowledge this is the only conference that has specifically focussed on these interrelated and very active topics of research. This book is not only the proceedings of this conference but is a tribute to one of the finest mathematicians of this century. The original impetus for the conference was to celebrate the80th birthday of Marshall Hall. With his untimely death on July4, 1990, the meeting became a memorial conference to honour the man and his lifetime contribution to mathematics.

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

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