# Hall, Philip

# HALL, PHILIP

(*b*. London, England, 11 April 1904; *d*. Cambridge, England, 30 December 1982)

*mathematics, algebra, group theory*.

Hall was an algebraist who worked in Cambridge for nearly fifty years, beginning in 1927, when he moved smoothly from being a student to becoming a research worker. He was enormously influential both through his writings and through his research students. His main work was in group theory, in which he made fundamental discoveries, particularly about soluble groups.

**Origins and Education** . Hall’s parents were George Hall and Mary Laura Sayers (1872–1965). They were not married and the father disappeared soon after Hall’s birth. Philip was educated at a local primary school until he won a scholarship to Christ’s Hospital, a foundation dedicated to the free education of orphans (which at the time meant fatherless children), which he joined in May 1915. From Christ’s Hospital, where his mathematical talents had become evident, he won a scholarship and sufficient financial support to enable him to enter King’s College, Cambridge, in October 1922. Hall was one of the Wranglers in the Mathematical Tripos of 1925 and took his BA that same year. As was common at the time, he did not study for a doctorate. The Cambridge undergraduate syllabus of the 1920s was rich in analysis, geometry, and applied mathematics but contained little algebra. There were lectures on group theory by Henry F. Baker and by F. P. White, however, which Hall attended, and he was encouraged by Arthur Berry, an assistant tutor at King’s College, to read some of the works of William Burnside, especially the book *Theory of Groups of Finite Order*(2nd ed., Cambridge, 1911) and some of his later papers, and questions on this material were set for him in the Tripos examination.

**Career** . The first year after graduation Hall remained in Cambridge, learning languages, competing unsuccessfully in the civil service examination and, presumably, thinking about finite abelian groups. He submitted a dissertation on that subject, which, in spite of the fact that it was unfinished, won him a fellowship at King’s College and had great influence on later writers. For the first few months of 1927 he worked as a research assistant to Karl Pearson in London, but statistics did not suit him and he returned to Cambridge and to group theory. Some years later, when Hall was elected to the Royal Society in 1942, he wrote:

The aim of my researches has been to a very considerable extent that of extending and completing in certain directions the work of Burnside. I asked Burnside’s advice on topics of group-theory which would be worth investigation & received a postcard in reply containing valuable suggestions as to what would be worth-while problems. This was in 1927 and shortly afterwards Burnside died [on 21 August 1927]. I never met him, but he has been the greatest influence on my ways of thinking. (Gruenberg and Roseblade, 1988, p. 8)

In Cambridge he was appointed university lecturer in 1933, promoted to reader in 1949, and elected Sadleirian Professor in 1953, but his primary allegiance was always to King’s College, where he remained a Fellow until his death and taught undergraduates for eight hours per week until he became professor. From September 1941 until the end of the World War II he had leave of absence from Cambridge to work in the Government Code and Cypher School at Bletchley Park, where he contributed to the decoding of Italian and Japanese material.

Hall was an uncommonly successful supervisor of research students. Although a little reclusive, he would offer students as much time and genial companionship— and mathematical guidance and advice—as they wished for. He guided Garrett Birkhoff in universal algebra and lattice theory during the academic year 1932–1933 and three (Bernhard H. [B. H.] Neumann, Paul Cohn, and James A. [J. A.] Green) of the twenty-nine students he supervised were, in due course, elected Fellows of the Royal Society. His influence was enormous, not only on their mathematics but also on their personality: several of his students acquired handwriting that was almost indistinguishable from his; several of them acquired one of the curious characteristics of his lecturing—that, when explaining the mathematics he would often be looking down at the floor to his left while pointing to the relevant formula on the blackboard behind him to his right.

Hall’s distinction was recognized in many ways. He was elected Fellow of the Royal Society in 1942 and awarded its Sylvester Medal in 1961. He served as president of the London Mathematical Society from 1955 to 1957 (he had served as one of its honorary secretaries (from 1938 to 1941 and from 1945 to 1948) and was awarded its Senior Berwick Prize in 1958 and its De Morgan Medal, the greatest mark of distinction the Society can offer, in 1965. He was awarded honorary doctorates by the universities of Tübingen (1963) and Warwick (1977), and he was elected to an Honorary Fellowship of Jesus College, Cambridge in 1976.

**Mathematics** . Hall’s major contributions were to five areas of algebra. In 1928 he published a short paper showing how the Sylow Theorems might be significantly extended for finite soluble groups. In 1872 Ludvig Sylow had published theorems guaranteeing *inter alia* that in a finite group *G* of order *p ^{a} m*, where

*p*is a prime number that does not divide

*m*, any subgroup whose order is a power of

*p*will be contained in a subgroup of order

*p*; moreover, all subgroups of order

^{a}*p*must be conjugate to each other. What Hall proved was that if

^{a}*G*is soluble and its order is

*mn*, where

*m*and

*n*are co-prime, then any subgroup whose order divides

*m*is contained in a subgroup of order

*m*(such subgroups were later known as Hall subgroups), and moreover all the subgroups of order

*m*are conjugate to each other.

Nine years later he published an even shorter paper proving a converse: If a finite group *G* has Hall subgroups of all possible orders then it must be soluble. These theorems, together with the theory of Sylow systems and system normalisers that he developed in 1937, became the basis for an extensive theory of finite soluble groups that developed particularly strongly in the 1960s and 1970s. The second theorem, as a characterization theorem for soluble groups, played a major part in the work of John Thompson and others working on the search for simple groups from 1960 to 1980 and beyond. Related, but in many ways very different, work on the *p*-length of *p*-soluble groups, published jointly with Graham Higman in 1956, had a great influence not only in providing tools for the search for finite simple groups but also in attacks on the Burnside Problem about groups of finite exponent. Hall rarely collaborated. In this case what happened was that Higman submitted a paper about groups of exponent 6 to the London Mathematical Society, and Hall, who had been appointed referee, recognized that the methods could be generalized far beyond where Higman had taken them, so they carried out the generalization together.

Hall contributed greatly to the theory of finite *p*-groups, that is, groups of prime-power order. In a long and much-cited article published in 1934, he extended some late work of William Burnside, establishing the modern theory of commutators and the lower central series in these groups. This was later extended to general nilpotent groups in lectures given in Canada in 1957. His theory of isoclinism of *p*-groups provides one of the main tools for classifying them—indeed essentially the only successful tool until the advent of co-class theory in the early 1980s.

Papers published in 1951, 1959, and 1961 opened up the theory of infinite soluble groups. Taking Hilbert’s Theorems about Noetherian rings and generalizing them to group-rings of finitely generated nilpotent groups, and treating the last term *L* of the derived series of a soluble group *G* as a module over the group-ring of the quotient group *G/L*, Hall was able to prove that finitely generated abelian-by-nilpotent groups satisfy the ascending chain condition for normal subgroups and are residually finite, and much more besides. Many of his students developed this line of thinking into an extensive area of algebra.

Hall’s interest in finite abelian groups, which earned him his fellowship of King’s College in 1927, has led to the eponymous Hall Algebra, which plays a large part in representation theory and the modern theory of symmetric functions. Hall’s only publication on the subject was a brief survey that summarized lectures given at the Fourth Canadian Mathematical Congress in 1957, but his ideas have been disseminated through lectures and through the exposition and extension of his work by Ian G. Macdonald and others.

This necessarily brief account of Hall’s work and influence omits mention of much of his work on aspects of infinite group theory—his studies of locally finite groups, of characteristically simple groups, of simple groups, and much more besides. He published relatively few papers, but every paper made a significant point and was greatly influential in one way or another. He was the most cited group theorist of the twentieth century.

## BIBLIOGRAPHY

### WORK BY HALL

Gruenberg, K. W. and J. E. Roseblade, eds. *The Collected Works of Philip Hall*. Oxford: Clarendon Press, 1988. Contains Hall’s thirty-four research articles.

### OTHER SOURCES

Gruenberg, K. W., and J. E. Roseblade. *Group Theory: Essays for Philip Hall*. London: Academic Press, 1984.

Macdonald, I. G.*Symmetric Functions and Hall Polynomials* . 2nd ed. Oxford: Oxford University Press, 1995.

Roseblade, J. E., J. A. Green, and J. G. Thompson. “Philip Hall.” *Bulletin of the London Mathematical Society* 16 (1984): 603–626. Obituary. Also published in *Biographical Memoirs of Fellows of the Royal Society* 30 (1984): 603–626, and reprinted in the *Collected Works of Philip Hall*, 1988.

*Peter Neumann*

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**Hall, Philip**