# Philip Hall

### Quick Info

Born
11 April 1904
Died
30 December 1982
Cambridge, England

Summary
Philip Hall was the main impetus behind the British school of group theory and the growth of group theory to be one of the major mathematical topics of the 20th Century was largely due to him.

### Biography

Philip Hall's father was George Hall and his mother was Mary Laura Sayers. They were not married and George left without making any provision for Laura or their newly born son Philip. Laura called herself Mrs Hall and she brought Philip up in her parents home in Hampstead, London, until he was seven years old. At the time Philip was born Laura was employed as dressmaker as was her twin sister Lois, and also her two elder sisters, who were also twins, Ada and Ethel. In 1909, while living in his grandfather Joseph Sayers' house, Philip entered New End Primary School.

In 1910 Laura, along with her three sisters, bought a house in Well Walk which they ran as a boarding house. Philip attended Sunday School and was baptised in 1911. He excelled at his Primary School and in 1915 he won a scholarship to Christ's Hospital West Horsham. This was a boarding school for children of ability whose parents were not able to afford the normal boarding school fees. Hall entered Christ's Hospital in May 1915 at age eleven. Although these were the years of World War I, Hall was too young to be affected by the war, other than serving in the Officer's Training Corps.

It was at Christ's Hospital that Hall came to love mathematics. He was fortunate to have teachers who were both accomplished mathematicians and also able to transmit their enjoyment of the subject. It was not only mathematics in which he excelled, winning the Gold Medal in his final year, but also in English winning a medal for the best English essay. Hall was House Captain in 1921-22, his final year at Christ's Hospital, and his popularity says much about his character [4]:-
He was kind and helpful to the younger boys, who respected and admired him; this was remarkable in those days for he was neither an extroverted leader nor any kind of athlete. He played rugby for his House as a rather clumsy forward, but gave up cricket, being content for a number of years to be official scorer for the school First Eleven.
His fellow school pupils recollected that at school he was [4]:-
... likeable and cheerful, with a sense of humour, gentle and reserved.
Hall went up to King's College Cambridge in October 1922 having won an Open Foundation Scholarship in December 1921. He wrote in his first letter home (see for example [4]):-
I am getting on beautifully and enjoying it very much; there are such opportunities of learning that it is about all you can do to make the most of them. ... I have made the acquaintance of Mr Littlewood and Mr Pollard, the two most progressive of the mathematicians here, so I am going on fairly well for a start.
The 'Mr Littlewood' that Hall referred to is Dudley Littlewood. However, he fails to mention in this letter one other extremely promising mathematician in his year at Cambridge, namely William Hodge. Among his teachers at Cambridge were Hobson, the Sadleirian professor, and Baker, the Lowndean professor of Astronomy and Geometry. Richmond was also on the staff when Hall arrived in Cambridge, but he retired in 1923.

Hall's interest in group theory came from Burnside's book which he was encouraged to read by Arthur Berry, the Assistant Tutor in Mathematics at King's College. Hall later wrote:-
I began with Berry's encouragement to study the works of William Burnside, especially his magnificent treatise on the "Theory of Groups" and some of his later papers.
Hall offered parts of that book for examination in the Tripos and gave a proof that no group of order $p^{n}, n > 1$, can be simple. He graduated with a B.A. in 1925 and was elected to an Open Senior Foundation Scholarship which supported him for a further year at King's College. It was a year in which Hall wondered about his future, unsure whether to try for an academic career or not. He sat the Civil Service Examination in June 1926 which, if he had been successful, would have given him a fast route to the Administrative Grades. Fortunately for mathematics, and particularly group theory, he was not successful. He did spend some time on learning languages during this year; he spent the summer of 1925 in Italy learning Italian and studied German in London in March 1926.

In October 1926 Hall submitted an essay The Isomorphisms of Abelian Groups as his attempt to gain a Fellowship. It shows many signs of having been written hurriedly, even to the extent that it ends in the middle of a proof! It is fairly clear that Hall only made the decision to try for an academic career after much thought and late on in terms of writing up his dissertation. Despite its deficiencies, it shows that already Hall was way ahead of his time in his approach to group theory and certainly nobody at Cambridge could have been in a position to properly evaluate the work. It considers subgroups of $PGL(2,\mathbb{C})$ and, among other results, proves that a group of order $p^{n}, n > 1$, in which every characteristic abelian subgroup has order $p$, is the central product of non-abelian groups of order $p^{3}$. John Thompson writes in [4] that the dissertation:-
... suffers from unwise use of the word 'obvious', a trait common to the young, but not always confined there. It is a trait which Hall did not retain.
Despite having written the work hurriedly, his quality shone through and Hall was elected to a Fellowship at King's College in March 1927. By that time he was already working as a research assistant to Karl Pearson in University College, London. He took up this post in January 1927 and his first published papers are on the theory of correlation. However, he found his main task of computing tables for the Incomplete Beta Function less than inspiring.

Hall wrote to Burnside in the summer of 1927 and, in 1942 he described this:-
I asked Burnside's advice on topics of group theory which would be worth investigation and received a postcard in reply containing valuable suggestions as to worth-while problems. ... shortly afterwards Burnside died. I never met him, but he has been the greatest influence on my ways of thinking.
Returning to Cambridge in September 1927 to take up the Fellowship at King's he made an important discovery in group theory, generalising the Sylow theorems for finite soluble groups to prove what are now called Hall's theorems. These fundamentally important results were published in A note on soluble groups in the Journal of the London Mathematical Society in 1928.

There is then a rather surprising gap in Hall's publication record. Here are his own words, written fifty years later:-
My Fellowship at King's had been renewed in 1930 but, sometime in 1931 it was intimated to me that a second renewal would be unlikely unless I showed signs of mathematical life; before then I had only produced one short note in 1928, so there was some justification in their warning and I obviously had to make a bit of an effort.
Hall certainly made 'a bit of an effort' for in 1932 he wrote what is perhaps his most famous paper A contribution to the theory of groups of prime power order. It is a beautiful paper which is one of the fundamental sources of modern group theory. In it, in addition to its main aims of developing the theory of regular $p$-groups, Hall introduces the commutator calculus, commutator collection, and the connection between $p$-groups and Lie rings. Not only did he get his Fellowship renewed but in 1933 he was appointed as a Lecturer at Cambridge.

In June 1939 Hall gave a series of lectures at a small meeting at the Mathematical Institute in Göttingen. Four of Hall's lectures were published as separate papers in Crelle's Journal. These papers are Verbal and marginal subgroups, The classification of prime-power groups, On groups of automorphisms, and The construction of soluble groups all of which appear in volume 182 published in 1940. In The classification of prime-power groups Hall introduces an equivalence relation called isoclinism to aid the classification of prime power groups. This important concept continues to play a major role. We should note that Hall was criticised for going to Germany at this difficult time but defended his actions saying:-
... the German mathematicians ... [are] as little responsible for the present situation (and probably enjoy it as little) as you or I do.
During World War II he made an important contribution with his work at the Code and Cypher School at Bletchley Park where he began work in September 1941. In particular he worked on Italian ciphers, then on Japanese ciphers learning about 1500 Japanese characters to help him in this task. During these war years he lived with his mother in Little Gaddesden where she had moved with her elder sister Ada at the beginning of the war in 1939 to be away from London. This meant that he had to travel about 20 miles to Bletchley Park each day and he made the 40 mile round trip partly by train and partly by motorcycle.

Hall returned to King's College Cambridge in July 1945. In 1946 he wrote letters to the authorities supporting Hasse's reinstatement, and also wrote encouraging letters to Hasse who had shown great kindness to Hall in 1939. Hall was promoted to Reader at Cambridge in 1949, then in 1953, after Mordell retired from the Sadleirian Chair, Hall was appointed to succeed him. In 1955 he was one of the main speakers at the Edinburgh Mathematical Colloquium in St Andrews where he gave five lectures on Symmetric Functions in the Theory of Groups. His picture at this colloquium is one of those in theset above.

Before giving his lectures he wrote to Edge saying:-
The subject I have in mind is symmetric functions, in relation to various branches of the theory of groups. I think I can find something to say on that which will not be too trite.
In particular he spoke about partitions and their connection to representation theory:-
... whenever in mathematics you meet with partitions, you have only to turn over the stone or lift up the bark, and you will, almost infallibly, find symmetric functions underneath. More precisely, if we have a class of mathematical objects which in a natural and significant way can be placed in one-to-one correspondence with the partitions, we must expect the internal structure of these objects and their relations to one another to involve sooner or later ... the algebra of symmetric functions.
In 1956 Hall published, jointly with Graham Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's problem. This is a paper of major importance as was seen by Baer when he wrote a review saying that he could:-
... do no more than just indicate the wealth of material contained in this investigation.
The paper has indeed proved highly influential and much of the rapid development of group theory in the 1960s was built on this foundation. In August 1957 Hall gave a series of lectures at the Canadian Mathematical Congress Summer Seminar in Edmonton, Canada, on nilpotent groups which have had great influence ever since.

His major contribution to infinite groups is seen in highly significant papers of 1952, 1959 and 1961. The ideas of these papers continue to be one of the main areas of group theory research. For example The Frattini subgroups of finitely generated groups is the important paper on infinite groups which he published in 1961. In it Hall considers many different classes of groups and investigates whether the Frattini subgroup of groups in these classes needs to be nilpotent.

In On non-strictly simple groups published in 1963 Hall established the existence of simple groups which were the infinite union of a chain of subgroups, each normal in the next. The paper, like so many of Hall's papers, introduces important ideas which are widely applicable. Karl Gruenberg explains some further features of this paper:-
Besides containing a discussion of the possible order types of abelian series in simple groups, the paper also presents an extremely informative survey of the inter-relations that are known or conjectured to exist between the various classes of generalized soluble groups. This discussion is kept concise by the use of an elegant calculus of closure operations on group properties.
Hall received many honours for his work. He was elected to the Royal Society in 1942, then he was awarded its Sylvester Medal in 1961:-
... in recognition of his distinguished researches in algebra.
Hall was a great supporter of the London Mathematical Society, and he was awarded its Senior Berwick Prize (1958) and the De Morgan Medal and Larmor Prize in 1965. He was elected President of the London Mathematical Society in 1955 and served the Society in this capacity until 1957. He delivered his Presidential address on 21 of November 1957 on Some word-problems. In this talk Hall spoke about word problems in general and specifically mentioned word problems for groups, semigroups, and cancellative semigroups. He introduced the idea of a normal form which he used in the solution of the word problem for Lie rings and also for nilpotent groups. He ended his talk with these words:-
Problems such as these still seem to present a formidable challenge to the ingenuity of algebraists. In spite of, or perhaps because of, their relatively concrete and particular character, they appear, to me at least, to offer an amiable alternative to the ever popular pursuit of abstractions.
The collected works of Philip Hall [2] were published in 1988. A McIver in a review wrote:-
This beautiful book consists of almost fifty years of publications by one of the greatest mathematicians of this century. ... Hall's elegant works (both in content and exposition) are allowed to speak for themselves ... however, the tremendous impact which his research has had on algebra is discussed... . The reader glimpses a little of his character: his universal kindness and his invigorating enthusiasm both for mathematics and the world in general. ... Altogether we are presented with an all-around picture of a most remarkable mathematician.
We have made various comments about Hall's character in this article, but we should end by making a few more. He had a deep love of poetry which he recited beautifully in English, Italian or Japanese. He also loved music, art, flowers, and country walks. However, he was a rather shy man who avoided large gatherings and was only really happy in company when he was with one or two friends. When Olga Taussky-Todd accused him of being the worst recluse in Cambridge, Hall replied "No, Turing is worse"! He had an incredibly broad knowledge, not only of mathematics but, it seemed, on almost any subject [1]:-
Hall's range of knowledge was extraordinary, embracing anything from agriculture to poetry, ... combined with his complete integrity, high intellectual standards and sound judgement ...
Although a man of a few words, his comments were always significant. His modesty was clear when you spoke to him or heard him lecture as I [EFR] had the good fortune to do on several occasions. Roseblade, one of Hall's research students, writes in [4]:-
His students loved him and he them. Writing so lucidly and elegantly himself, he must have found painful much of what they first wrote; but whenever he had strong criticism to make of their work, he always found a way to soften the blow and never failed to suggest effective improvements. Nor did he abandon them when they had completed their dissertations; he wrote them helpful and stimulating letters, often very long and always by hand. ... He was a wonderful person; gentle, amused, kind, and the soul of integrity.

### References (show)

1. Obituary in The Times
2. P Hall, The collected works of Philip Hall (New York, 1988).
3. J L Brenner, Philip Hall - A famous mathematician, Pi Mu Epsilon Journal 9 (1990), 110-111.
4. J A Green, J E Roseblade and J G Thompson, Philip Hall, Bull. London Math. Soc. 16 (6) (1984), 603-626.

### Additional Resources (show)

Other pages about Philip Hall:

Other websites about Philip Hall:

### Honours (show)

Honours awarded to Philip Hall

### Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update October 2003