by J. E. Roseblade, rev.

© Oxford University Press 2004 All rights reserved

**Hall, Philip** (1904-1982), mathematician, was born on 11 April 1904 in Hampstead, the natural son of George Hall and Mary Laura Sayers, dressmaker, of Hampstead, daughter of Joseph Sayers, gardener, of Balcombe. George Hall disappeared from his life soon afterwards and Philip was brought up by his mother. He was educated at Christ's Hospital and, as a scholar (1922-5) and senior scholar (1925-6), at King's College, Cambridge. He was in the first class in part one (1923) and a wrangler in part two (1925) of the mathematical tripos. In 1927 he was elected to a fellowship at King's, which he held for the rest of his life. He was appointed university lecturer in 1933. From 1941 to 1945 he worked at the Government Code and Cypher School at Bletchley Park, first on Italian ciphers and later on the Japanese diplomatic ciphers. He was reader in algebra at Cambridge from 1949 and Sadleirian professor of pure mathematics from 1953 until 1967, when he elected to retire.

While still an undergraduate Hall began to study the works on group theory of William Burnside, whom he never met but who was the greatest influence on his ways of thinking. In *A Note on Soluble Groups* (1928), he proved a result as important for the theory of finite soluble groups as Sylow's theorem of 1872 is for finite groups in general, that if *G* is a soluble group of order *mn,* where *m* and *n* are coprime, then every subgroup of *G* whose order divides *m* is contained in some subgroup of order m, and these subgroups of order *m* are all conjugate in G. Ten years later he characterized soluble groups by such arithmetic properties, and went on to develop a general theory of finite soluble groups. Hall's most influential pre-war paper was *A Contribution to the Theory of Groups of Prime-Power Order* (1934), in which he discovered many features which underlie the structure of the most general p-group, initiated the theory of regular p-groups, laid down the basic laws of the commutator calculus, and revealed one of the links connecting the study of groups with that of Lie rings. In 1935 he proved a fundamental combinatorial result known as the marriage theorem, which became part of mathematical folklore.

Between 1940 and 1954 Hall published no mathematics. His work in the mid-1950s on theorems like Sylow's and, in collaboration with Graham Higman, on the p-lengths of p-soluble groups was indispensable for the great achievements in finite group theory of the 1960s. He also studied combinatorial questions arising from the appearance of partitions in different parts of group theory and invented an important and elegant algebra of partitions. He made many contributions to the theory of infinite groups. Of seminal importance was his systematic investigation between 1954 and 1961 of certain finiteness conditions in soluble groups, which, for example, initiated the representation theory of polycyclic groups. In 1959 he constructed a universal locally finite simple group and in 1963 a non-strictly simple group, and in 1974 he proved very general theorems about embedding groups in simple groups.

Both through his own work and through that of his students, to whom he was always most generous with his ideas, Hall exercised a profound influence on English mathematics, which was felt throughout the mathematical world.

He had unusually wide interests in both the sciences and the humanities, with an encyclopaedic knowledge and prodigious memory. For many years when doing mathematics he would smoke one cigarette after another, but, to test his will, used, from time to time and for predetermined periods, to give them up. Hall was unmarried and for much of his life lived alone, always studious and caring nothing for hot water or central heating. He had little liking for large gatherings or formal occasions, but behind his shyness lay a particularly friendly disposition, and when he was with friends he was the best company in the world. He had an extensive but discriminating love of poetry, which he spoke beautifully, not only in English. He enjoyed music and art, flowers, and country walks. He was gentle, amused, kind, and the soul of integrity.

Hall was elected FRS in 1942 and received the Sylvester medal in 1961. He was president of the London Mathematical Society (1955-7) and was awarded both the senior Berwick prize (1958) and the De Morgan medal and Larmor prize (1965). He had honorary doctorates from Tübingen (1963) and Warwick (1977). From 1976 he was an honorary fellow of Jesus College, Cambridge. Hall died at Addenbrooke's Hospital, Trumpington Street, Cambridge, on 30 December 1982 and his ashes were interred in his mother's grave in Impington churchyard near Cambridge. He left half his residuary estate to the National Trust.

J. E. ROSEBLADE, *rev.*

**Sources **

J. A. Green, J. E. Roseblade, and J. G. Thompson, *Memoirs FRS,* 30 (1984), 251-79

personal knowledge (1990)

*WWW*

*CGPLA Eng. & Wales* (1983)

d. cert.

**Likenesses **

photograph, repro. in Green, Roseblade, and Thompson, *Memoirs FRS*

**Wealth at death **

£158,600: probate, 18 April 1983, *CGPLA Eng. & Wales*

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