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Graham Higman was a highly talented mathematician whose principal field of research was group theory. He played a key role in the development of this field, and is regarded as one of the three most significant British group theorists of the 20th century, alongside William Burnside (FRS 1893) and Philip Hall (FRS 1951). He studied at Oxford, gaining a doctorate under the supervision of Henry Whitehead (FRS 1944). After working for the Meteorological Office during the Second World War, he was appointed to a position at the University of Manchester in 1946, and subsequently at Oxford. In 1958 he was elected FRS, and in 1960 he was appointed to the Waynflete professorship in mathematics at Oxford, a position he held until his retirement in 1984. His research was significant and influential. Highlights include his work on embeddings of groups, including 'HNN-extensions'and their application in the famous Higman Embedding Theorem, and his constructions of a finitely-presented group that is isomorphic to a proper factor of itself, and a finitely-presented infinite simple group. Some of his work also played an important role in the proof of the Odd Order Theorem and a solution to the restricted Burnside problem. He and his wife Ivah were dedicated Methodists, and Graham was a keen bird-watcher. He was also well known for his view that pure mathematics should be studied for its beauty and its life-affirming properties, for the non-standard yet highly supportive way he supervised his students and for his somewhat mischievous sense of humour.
Early life and education
Graham Higman was born on 19 January 1917, in Louth, Lincolnshire, as the second son of the Reverend Joseph Higman and Susan Mary Higman (née Ellis). His father was a United Methodist minister, and the family moved from Lincolnshire to London (1918-1924), and then to Long Eaton (1924-1929) and to Plymouth, where Graham was educated at Sutton High School for Boys, a grammar school in Plymouth.
In 1934 he won a natural sciences scholarship to study at Balliol College, Oxford. He chose Balliol because his elder brother Bryan had studied there, but deciding to distinguish himself from Bryan (who studied chemistry), he chose to study mathematics.
Graham's tutor was the great pure mathematician Henry Whitehead (FRS 1944), a leader in the fields of algebraic and geometric topology, who was later appointed to the Waynflete professorship in pure mathematics. Whitehead initially assumed that Graham's natural sciences background would make him study applied mathematics, or take mathematics courses only to support his science studies, but it soon became clear that Graham's principal interests lay in pure mathematics. In fact, in 1936 he co-founded (with Henry Whitehead and a fellow student, Jack de Wet) the Oxford University Invariant Society, which was dedicated to promotion of interest in mathematics. The opening lecture was given by G. H. Hardy (FRS 1910), on 'Round numbers'. This society is also known as 'The Invariants'(a name chosen at random by Graham from the titles of the books on Henry Whitehead's bookshelf), and it still exists today.
Unsurprisingly, Graham completed his degree with a First, and after taking special-topic courses on group theory and differential geometry, he was awarded an MA. He then stayed on in Oxford to study for a doctorate, supervised by Henry Whitehead, working on the units of group-rings because of a geometric question that had arisen in algebraic topology. Among other things in his thesis, he classified group-rings over the rational numbers without non-trivial units. He was awarded a DPhil in 1941 (at the age of 24).
The war years
Graham married Ivah Treleaven in 1941, and over the next few years they had five sons and a daughter.
Following his doctoral studies, Graham spent a year at the University of Cambridge, where he was strongly influenced by Philip Hall (FRS 1942), and also met Max Newman (FRS 1939), whose interest in the interaction between group theory and logic had a lasting influence on him. (Newman worked at Bletchley Park during the Second World War, on the Colossus machine.)
While his brother served in the RAF during the Second World War, Graham registered as a conscientious objector, but he interrupted his academic career to serve in the Meteorological Office--first in Lincolnshire, and then in Northern Ireland and Gibraltar. At the end of the war, he applied for a permanent position in the Meteorological Office, but after being asked at an interview why he had not chosen to enter the academic world and then being offered the position, he decided to turn down the offer and pursue an academic career.
Academic career
Graham was offered a position at Durham University, but he turned this down before accepting an offer from the University of Manchester, where Max Newman was building the mathematics department along lines that were then rare in British universities, modelled on his experience at Bletchley Park. In 1946, Graham was appointed as a lecturer at Manchester, in an intensely research-oriented group of young mathematicians, including Alan Turing (FRS 1951). Walter Ledermann was also appointed at Manchester in 1946, and then Bernhard Neumann (FRS 1959) arrived in 1948.
Graham engaged in a highly successful (and often competitive) collaboration with Bernhard Neumann and his wife Hanna Neumann, some of the results of which will be described below. As Graham himself described to his student Brian Stewart, he and the Neumanns would sometimes part on a Friday for the weekend, stuck on some part of their work, and each of them would vie to be the one who produced the breakthrough on the Monday morning.
Graham was ambitious, and began to apply for professorships at quite an early stage of his career, but Henry Whitehead encouraged him to return to Oxford rather than to seek a chair at a lower-ranked institution. In 1955 Graham was appointed as a lecturer in mathematics in Oxford, and very soon afterwards he was promoted to reader. In 1958 he was honoured with election as a Fellow of the Royal Society, and also became a senior research fellow at his former college, Balliol. Then, following Whitehead's premature death, Graham was appointed to the Waynflete professorship in mathematics, ahead of the younger Michael Atiyah (FRS 1962). This involved a move to a fellowship at Magdalen College, and he held both positions from 1960 until his retirement in 1984.
As Waynflete professor, Graham had numerous doctoral students (many of whom are listed later in this memoir). His approach to supervision was not unusual for the time, but was certainly different from current expectations. He disliked scheduling regular meetings, and preferred that his students would attend colloquia and advanced classes, not just those he conducted himself, appear for tea in the Mathematical Institute's common room on a regular basis (to engage in mathematical conversations), and come to see him only when they had something important to say or ask; and when that happened, he would put aside his own work in order to help the student, offering many ideas and explanations (usually sketched on a board), and yet he would not seek credit for those ideas, let alone eventual joint authorship.
Remarkably, on at least two occasions Graham arranged for two contemporaneous students to prove complementary halves of what became a joint theorem, including one by Sheila Oates and Martin Powell (on identical relations in finite groups), and another by Steve Smith and Peter Tyrer (on finite groups with a certain Sylow normaliser).
Graham's approach helped to develop a large degree of independence and self-reliance in most (but not all) of his students. One notable success is highlighted by an account of him telling a student: 'When I asked you a question, I didn't expect you to prove a theorem.'Sadly, a small number of students (and a few colleagues) found him intimidating, possibly because of his high academic standing and reputation, and some needed quite a lot of help, or a change of topic or supervisor. It has been reported that he once remarked that there were three types of student: those who wrote their own thesis; those for whom he wrote their thesis and they understood it; and those for whom he wrote their thesis, but they did not. Overall, he was generous with his time and help, on the basis of need.
His advanced classes covered a wide range of topics, and often contained his most recent thinking--sometimes worked out only an hour or so before the class--and occasionally he would arrive late if some detail had not quite worked out in the way that he expected. As students found in his undergraduate lectures as well, it was quite a privilege to learn from him, even if it took some time to assemble and understand the details, not all of which were given in full.
In terms of professional service, Graham co-founded the Oxford University Invariant Society in 1936 (as reported earlier), and also he served as the fifty-second president of the London Mathematical Society, from 1965 to 1967. Perhaps his greatest professional service contribution, however, was as founding editor of the Journal of Algebra, which grew to become the leading journal in algebra in a relatively short period. Graham served as its editor-in-chief from 1964 until his retirement in 1984, although his filing system was sub-optimal and so some Oxford colleagues assisted with his editorial responsibilities towards the end.
He also served as head of the Mathematical Institute at Oxford, and as chairperson of the Faculty Board, and he did well to ensure that mathematics gained an appropriate share of resources at a time when the university was expanding. Additionally, he was one of the prime movers to establish Oxford's Honours School of Mathematics and Philosophy.
Graham was renowned for his disarming ability to ask simple and seemingly innocent questions that called for evidence in support of claims made at meetings. This sometimes had the effect of exposing the ambition of certain colleagues. When new positions were to be filled, he was against attempting to make appointments in particular fields that were deemed by some to be important, but was strongly in support of appointing people doing the best mathematics.
Graham made a number of visits to the USA over the course of his career, including one for the 1960/61 academic year in Chicago, at a time when an explosion of interest in finite simple groups had occurred following some outstanding work by John Thompson (FRS 1979) that significantly extended the methods of Hall and Higman (see later). Also, immediately after he retired from Oxford, Graham spent two years as George A. Miller visiting professor at the University of Illinois, from 1984 to 1986, and then spent a year travelling around the world, visiting former students and other colleagues.
Research field Graham Higman's principal field of research was group theory, which is a branch of algebra that grew out of the study of number systems, the roots of polynomial equations and geometry, and now forms the basis of the study of symmetry.
In an abstract setting, a group consists of a collection of elements that may be combined by a single operation subject to certain laws. The given operation could be addition (of numbers or matrices or polynomials, for example), or multiplication (of non-zero numbers, non-singular matrices or non-zero rational functions), or composition (of permutations, linear/geometric transformations or symmetries of an object), or one from a range of further possibilities.
The common properties of these operations were observed and abstracted by Évariste Galois, Augustin Louis Cauchy (ForMemRS), Arthur Cayley (FRS 1852), William Burnside FRS, Ferdinand Georg Frobenius and others in the 1800s, resulting in the establishment of the theory of groups, which is now one of the cornerstones of mathematics and continues to grow and develop strongly.
Group theory plays a fundamental role in many branches of mathematics, and has important and sometimes unexpected applications in a wide range of other fields. Applications in physics, chemistry and material science are widely known, for example in the study of crystal and molecular structures. Incidentally, the structure of the Buckminsterfullerene (or C60 molecule) was already known to group theorists as a Cayley graph for the alternating group on five points decades before the molecule was discovered, and known even earlier (owing to its symmetry) by Kepler and Archimedes. More recently, group theory has played a role in the construction of efficient networks (with very good 'broadcast'properties), and 'expander graphs'(sparse networks with very good connectivity properties).
A major highlight in the development of group theory was the classification of the finite simple groups, which are building blocks for all finite groups, akin to the chemical elements. This classification took place over a period of more than 25 years, involving the collaborative effort of over 100 mathematicians from around the world, published in journal articles taking up more than 10 000 journal pages, and essentially completed by 1980.
Refinements and clarifications of the long and highly complicated proof are ongoing, but, more importantly, the classification has resulted in many further significant discoveries, and much of the work involved in it has led to the rich development of closely related fields such as representation theory and geometric group theory.
Graham Higman played a key role in the development of group theory over the course of his career, and is regarded by many as one of the three most significant British group theorists of the 20th century, alongside William Burnside and Philip Hall. He once remarked that finite group theorists were the natural successors to classical geometers, but in fact he is well known for having made original, significant and influential contributions to the study of both finite groups and infinite groups, and across a wide range of topics, as described in the next section.
Like G. H. Hardy, Graham was unashamedly a pure mathematician. (Hardy paid little attention to applications, and yet his own field of number theory lies behind the advanced systems of encryption and other forms of information security we now use on a daily basis.) He strongly held the view that pure mathematics should be studied for its beauty and its life-affirming properties, rather than its utility.
In an interview in 1987 Graham said this:
Arguably, his most famous piece of work took place in collaboration with Bernhard and Hanna Neumann at Manchester, on embedding theorems for groups, published in a 1949 paper in the Journal of the London Mathematical Society (2). The principal outcome of this work was a construction for what is now known as an 'HNN extension'(based on the initial letters of the surnames of the authors), defined as follows.
Let be a finite or infinite group with presentation , meaning that is a generating set for and that is a set of defining relators for (expressed in terms of the members of ), and suppose that and are isomorphic subgroups of (which could be finite or infinite) and is an isomorphism between them, then the HNN-extension of relative to α is the group .
A key observation made and proved by the three authors is that the group is embedded as a subgroup of via an injective homomorphism. An important but non-obvious consequence is that two isomorphic subgroups and of a given group are always conjugate (by some element ) inside some 'overgroup' containing as a subgroup. The latter was the motivation for this construction, which now has several important applications across a wide range of fields. For example, in algebraic topology an HNN extension enables understanding of the fundamental group of a topological space that has been 'glued'back on itself via some mapping, in the same kind of way as free products with amalgamation do for gluing two spaces along a common connected subspace. Moreover, both of these constructions are basic building blocks in the Bass-Serre theory of groups acting on trees, and HNN-extensions are now used frequently in geometric group theory and their use has been extended in the study of Lie algebras.
Shortly after the HNN paper, Graham published two more very important papers in 1951: one giving an example of a finitely-presented group that is isomorphic to a proper factor of itself (3), and another giving his famous example of a finitely-generated infinite simple group (4). These papers accelerated the growth of his high reputation.
Also while still in Manchester, Graham undertook further successful research with Bernhard Neumann and other work on unrestricted free products and topological groups, and more significantly on questions that led to a celebrated joint paper with Philip Hall in 1956, on -soluble groups and reduction theorems for the Burnside problem (7). (The latter problem, posed by William Burnside in 1902, asked whether a finitely-generated group in which every element has finite order must be a finite group. It was solved in 1964 by Evgeny Golod and Igor Šafarevič, who provided a counter-example through their solution of the class field tower problem, showing that such towers can be infinite (Golod & Šafarevič 1964).)
The Hall-Higman paper (7) contributed ideas that have influenced finite group theory ever since it appeared. In particular, this paper included a reduction theorem for the restricted Burnside problem, which asked that if m and n are positive integers, and is any finite group generated by elements and the order of every element of is a divisor of , then is the order bounded by some constant depending only on and ? Hall and Higman essentially reduced this problem to looking only at groups of prime-power exponent, and their reduction played a vital part in Efim Zel'manov's positive solution to the restricted Burnside problem in the early 1990s (Zel'manov 1990, 1991).
Another theorem in the famous Hall-Higman paper (now known as the Hall-Higman Theorem) describes the possibilities for the minimal polynomial of an element of prime-power order in a representation of a -soluble group. More generally, this paper provided the basic tools with which Walter Feit and John Thompson were later able to resolve a conjecture of Burnside by proving the Odd Order Theorem, namely that every finite group of odd order is soluble (Feit & Thompson 1963). It has also been suggested that it is no accident that a remarkable but simple idea that lies at the heart of the Hall-Higman paper is numbered Lemma 1.2.3.
Graham used HNN-extensions in another of his most famous pieces of work, now known as the 'Higman Embedding Theorem', published in the Proceedings of the Royal Society of London in 1961 (11). This theorem states that a finitely-generated group can be embedded in a finitely-presented group if and only if it is recursively presented.
As a corollary, Graham proved that there exists a universal finitely-presented group that contains every finitely-presented group as a subgroup, up to isomorphism, and indeed the finitely-generated subgroups of are precisely the finitely-generated recursively presented groups, again up to isomorphism. The Higman Embedding Theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely-presented group with an algorithmically undecidable word problem. This work was of especially wide interest since Graham needed to work at the boundaries of logic to establish the concept of relations being 'recursively enumerable'. Also part of his proof of this theorem involved an ingenious argument often referred to as 'Higman's Rope Trick', in which he almost magically made an infinite set of relations disappear. Later in his career, Graham indicated that he viewed his Embedding Theorem as his greatest achievement.
He spent the 1960/61 academic year in Chicago, at a time when an explosion of interest in finite simple groups had occurred, following the PhD thesis by John Thompson that significantly extended the Hall-Higman methodology and proved the nilpotency of the kernel of a Frobenius group. (Also the Feit-Thompson Odd Order Theorem was proved that year.) Graham realised that these developments represented the medium-term future of group theory, but he did not become fully involved in them.
Being something of a maverick, Graham embarked on a programme of characterisations of finite simple groups. These provided suitable thesis topics for his students, but neither these nor his own work formed part of the core attack on the simple group classification. He did, however, use his knowledge and expertise to devise alternative constructions for some of the sporadic finite simple groups. One was an ingenious independent construction for the Higman-Sims group (HS) as the automorphism group of a 176-point block design (13). (Note here that the 'H' in HS denotes Donald Higman, possibly a distant relative.) Another was one for Janko's third simple group , using character theory to produce a finite presentation for the group, which was verified by John McKay using a coset enumeration on a computer (14), as with another for the Held simple group (unpublished).
Graham also devised a construction for Conway's third simple group from its regular two-graph. He did not publish this, or indeed anything on the theory of regular two-graphs he developed, but he let his student Don Taylor take over his advanced class to describe and extend the latter, which has garnered recent interest among people working at the interface between combinatorics and logic.
In the later part of his career, Graham developed a strong interest in Schreier coset graphs (which he called 'coset diagrams') and their use in constructing generating pairs for alternating and symmetric groups, and also in exhibiting actions of projective linear groups. Such diagrams were a key ingredient in my own doctoral thesis and a few subsequent publications, and I have fond memories of returning to my lodgings in Oxford one evening after a trip away and from a ride in a taxi past the Mathematical Institute, seeing Graham at his blackboard working on one. Indeed the frontispiece portrait of Graham in this memoir shows a coset diagram for the full (2, 5, 5) triangle group on 66 points.
Graham also published papers on a range of other topics, including the following: varieties of groups; Lie ring methods for finite nilpotent groups; enumerating -groups; generalised polygons; the algebraic structure of groups with soluble word problem and soluble order problem; the solubility of sets of equations over groups; criteria for non-simplicity of finite groups; homogeneous relations; countably free groups; groups with relatively free verbal quotients; and infinite permutation groups. Shortly before his retirement from Oxford in 1984, he gave a course on his recent work on existentially-closed groups, which he later published jointly with Elizabeth Scott (one of his last DPhil students) in the London Mathematical Society's Monograph Series, in 1988 (16).
Influential publications At the time of completing this memoir, and according to the American Mathematical Society's MathSciNet database (https://mathscinet.ams.org/mathscinet), Graham's published work has been cited 1810 times by 1738 authors. Both numbers appear to go up by about 10 per month.
His 12 most highly cited publications have citations (as recorded on the MathSciNet database at the end of January 2022) totalling 119, 160, 88, 284, 46, 135, 92, 64, 93, 95, 125 and 110, respectively. For the non-mathematical reader, it is worth noting that these are very high numbers of citations, for many reasons. Papers in mathematics tend to cite only the previous works with content that is actually needed for the current one, so the number of citations per paper is about five on average, and also the MathSciNet database counts only citations in publications covered by the database.
Mathematical genealogy Graham Higman's academic ancestry can be traced back via his doctoral supervisor Henry Whitehead to Oswald Veblen, Eliakim Hastings Moore, Hubert Anson Newton (ForMemRS), Michel Chasles (ForMemRS), Siméon Poisson FRS, Joseph Louis Lagrange FRS and Pierre Simon Laplace FRS, and further back to other renowned mathematicians including Leonhard Euler FRS, Gottfried von Leibniz FRS and Nicolaus Copernicus.
Graham supervised at least 50 doctoral students between the early 1950s and his retirement in the mid 1980s, including those listed in a supplement to this memoir. Well over half of these former students went on to forge academic careers of their own, in various places around the world, including Jonathan Alperin, Rosemary Bailey, Donald Barnes, David Cappitt, Christopher Clapham, Michael Collins, Marston Conder, Gabrielle Dickenson (Stoy), Leslie Fletcher, Jonathan Hall, Howard Hoare, John Howie, Geoffrey Howson, Ian Hughes, Peter Lambert, Ali Liggonah, John McDermott, Thomas McDonough, Glen Mullineux, Qaiser Mushtaq, Peter Neumann, Sheila Oates (Williams), Martin Powell, Alan Prince, Christopher Rowley, Peter Rowlinson, Elizabeth Scott, Stephen Smith, Brian Stewart, Don Taylor, Sean Tobin and George Wilmers. (I offer sincere apologies to any former students whose names I have missed.)
In turn, these people supervised students who now form part of a list of well over 840 academic descendants of Graham Higman (remarkably, with over half of those being descendants of Peter Neumann). Further details can be seen at the website for the Mathematical Genealogy Project (https://www.genealogy.math.ndsu.nodak.edu).
Family life and values, and sense of humour Both Graham and his wife Ivah were active members of the Methodist Church, particularly the Wesley Memorial Church in Oxford, and Wesley's hymns had a special place in Graham's heart. While still a student, Graham became a fully accredited Local Preacher in 1936. It has been reported that Graham's Methodism was far from old-fashioned, and that he welded his faith to an understanding of evolution, and thought deeply about the role of the after-death in religion, which he described as 'one of the murkier areas of Christianity'. (Also, however, I was told on more than one occasion over 40 years ago an amusing story about Graham's being asked to preach at a Christmas service, but then on the day, forgetting that it was actually Christmas.) When Ivah died in 1981, Graham gave the funeral address himself, and he continued as a lay preacher until 2001. The minister at his memorial service (in Oxford in May 2008) remarked that Graham's faith was unconventional.
Graham and Ivah were voluntary workers with the Samaritans, as well as avid 'greens'. One of their sons, Roger, achieved national eminence in 'Friends of the Earth', and a simple cardboard coffin was used for Graham's funeral.
Also Graham was a keen bird-watcher, renowned for sometimes arriving at the Oxford Mathematical Institute in a raincoat, with binoculars around his neck, and a field guide in a small back-pack, happily reporting on his latest sighting. Indeed when he visited Brazil after his retirement, he was asked by his hosts if there was anything they could get for him, and he suggested a field guide (to carry with him when out walking), but he was then presented with two very large leather-bound volumes which contained hundreds of beautiful photos but were completely impractical.
Graham's appearance was distinctive, with an impressive beard and bushy eye-brows. He claimed that the beard resulted from a broken razor during a family summer holiday in 1966. He could look amiable to some, but scary to others.
Many of those who knew him well, however, were aware of a mischievous sense of humour. Here I will give just a few examples.
In a review he wrote in 1966 on a book on universal algebra, he wrote this:
Some years later, at a short conference in Oxford he gave a lecture entitled 'The uselessness of logic', which naturally drew the attention of a large number of logicians and philosophers. In his lecture he showed that within a particular logical structure, it was possible to prove the existence of a universal object 𝒰 that would have every countable group as a homomorphic image, provided that 𝒰 had more than one element, but that within that logical structure, it was impossible to prove the latter requirement. Of course he had to field a large number of criticisms and questions at the end, but he was ready for all of them, with his enigmatic calm demeanour.
Here I will add one more thing that I witnessed first hand, not so much about his mischievous sense of humour but about an incident that tested his good humour. The weekly Algebra Seminar was held from 5pm to 6pm, after which the speaker and other participants would traditionally walk across St Giles for a drink and further discussion about the seminar topic at the 'Eagle and Child'(a favourite haunt of C. S. Lewis and J. R. R. Tolkien in their day), which opened at 6pm. One time, however, the visiting speaker finished very early and there were almost no questions, so the seminar ended over 20 minutes early. Poor Graham decided to walk home, somewhat annoyed, not only because advantage had not been taken of the full hour, but also because he did not want to be seen to be waiting around solely for the pub to open.
Closing remarks
Above all, aside from being an outstanding mathematician, Graham Higman was a proudly upright and yet humble individual.
His generosity of spirit is evident in his work for his church and the Samaritans, the stand he took against war, the love for his wife Ivah and their family, his concern for his students and his sense of humour. In a letter he sent to the Royal Society after being asked about aspects of this memoir (well in advance of it being written), he stated that he was uninterested in the personal aspects of his biography--but would be quite happy if his mathematical contributions were properly assessed--and also that he did not like any photograph of himself that he had seen, but he really liked his portrait by Norman Blamey and asked that it could be included. (Incidentally, one of the people who commissioned that portrait exclaimed when first shown the painting, 'That's the Graham we all know and love', imagining that Graham could be thinking 'I've proved a good theorem, now you lot do better!')
Graham's physical health deteriorated in his later years, but his mind did not. He spent the last few years of his life in a nursing home, but in 2007 he made a final (and well-dressed) public appearance at a conference in Oxford marking his ninetieth birthday (and the fiftieth anniversary of the Hall-Higman paper), much to everyone's surprise and pleasure.
It seems appropriate to conclude with this comment made about Graham by Don Manley in The Independent shortly after Graham's death (at the age of 91) in Oxford on 8 April 2008: 'In an age when successive governments want science and mathematics to justify themselves in terms of the national economy, this lovable and grumpy old man seemed to stand for something else that mattered much more.'
Awards and recognition
In addition to honorary degrees, Graham Higman was accorded the following honours for his outstanding work in mathematics:
1951 Morning Speaker, British Mathematical Colloquium
1957 Plenary Speaker, British Mathematical Colloquium
1958 Fellow of the Royal Society of London
1960-1984 Waynflete Professor of Mathematics, Magdalen College, Oxford
1962 Berwick Prize, London Mathematical Society
1965-1967 President of the London Mathematical Society
1974 De Morgan Medal, London Mathematical Society
1979 Sylvester Medal, Royal Society of London
Two others were the commissioning of the Norman Blamey portrait of Graham (shown as the frontispiece portrait of this memoir), and the establishment of the Higman Room in Oxford's Mathematical Institute to mark his retirement.
Acknowledgements
An invitation to write this memoir was initially made to Dr Peter Neumann (University of Oxford), one of Graham Higman's first doctoral students, but sadly Peter suffered a stroke and found he was unable to complete the task, and shortly afterwards died from Covid-19, in December 2020. I felt honoured to be invited to write the memoir in Peter's place, and I hope it comes close to what he would have written.
Next, I wish to acknowledge a variety of sources of information: the Bodleian Libraries online search facility (https://www.bodleian.ox.ac.uk/collections-and-resources/solo); the MacTutor History of Mathematics Archive (https://mathshistory.st-andrews.ac.uk); and the obituary articles that appeared in The Independent and The Telegraph in May 2008 and the Oxford Mathematical Institute Newsletter in the spring of 2009 (Anon. 2008; Collins 2008; Anon. 2009). The frontispiece portrait of Graham is reproduced with the kind permission of the Mathematical Institute, Oxford. It was painted by Norman Blamey in 1984, commissioned by former students and friends (who gave it to the institute) and it now hangs on a wall on the ground floor of the institute in the north wing of the Andrew Wiles Building at Oxford. I would also like to thank Stephen Blamey (son of the artist), Mike Giles (the current head of the Oxford Mathematical Institute) and Brian Stewart and Michael Collins (two of the portrait's commissioners) for arranging and endorsing permission to use it.
Last, but by no means least, I wish to heartily thank Rosemary Bailey, Martin Bridson, Peter Cameron, Gareth Jones and Brian Stewart for suggestions and additional points that greatly helped to improve this memoir.
Early life and education
Graham Higman was born on 19 January 1917, in Louth, Lincolnshire, as the second son of the Reverend Joseph Higman and Susan Mary Higman (née Ellis). His father was a United Methodist minister, and the family moved from Lincolnshire to London (1918-1924), and then to Long Eaton (1924-1929) and to Plymouth, where Graham was educated at Sutton High School for Boys, a grammar school in Plymouth.
In 1934 he won a natural sciences scholarship to study at Balliol College, Oxford. He chose Balliol because his elder brother Bryan had studied there, but deciding to distinguish himself from Bryan (who studied chemistry), he chose to study mathematics.
Graham's tutor was the great pure mathematician Henry Whitehead (FRS 1944), a leader in the fields of algebraic and geometric topology, who was later appointed to the Waynflete professorship in pure mathematics. Whitehead initially assumed that Graham's natural sciences background would make him study applied mathematics, or take mathematics courses only to support his science studies, but it soon became clear that Graham's principal interests lay in pure mathematics. In fact, in 1936 he co-founded (with Henry Whitehead and a fellow student, Jack de Wet) the Oxford University Invariant Society, which was dedicated to promotion of interest in mathematics. The opening lecture was given by G. H. Hardy (FRS 1910), on 'Round numbers'. This society is also known as 'The Invariants'(a name chosen at random by Graham from the titles of the books on Henry Whitehead's bookshelf), and it still exists today.
Unsurprisingly, Graham completed his degree with a First, and after taking special-topic courses on group theory and differential geometry, he was awarded an MA. He then stayed on in Oxford to study for a doctorate, supervised by Henry Whitehead, working on the units of group-rings because of a geometric question that had arisen in algebraic topology. Among other things in his thesis, he classified group-rings over the rational numbers without non-trivial units. He was awarded a DPhil in 1941 (at the age of 24).
The war years
Graham married Ivah Treleaven in 1941, and over the next few years they had five sons and a daughter.
Following his doctoral studies, Graham spent a year at the University of Cambridge, where he was strongly influenced by Philip Hall (FRS 1942), and also met Max Newman (FRS 1939), whose interest in the interaction between group theory and logic had a lasting influence on him. (Newman worked at Bletchley Park during the Second World War, on the Colossus machine.)
While his brother served in the RAF during the Second World War, Graham registered as a conscientious objector, but he interrupted his academic career to serve in the Meteorological Office--first in Lincolnshire, and then in Northern Ireland and Gibraltar. At the end of the war, he applied for a permanent position in the Meteorological Office, but after being asked at an interview why he had not chosen to enter the academic world and then being offered the position, he decided to turn down the offer and pursue an academic career.
Academic career
Graham was offered a position at Durham University, but he turned this down before accepting an offer from the University of Manchester, where Max Newman was building the mathematics department along lines that were then rare in British universities, modelled on his experience at Bletchley Park. In 1946, Graham was appointed as a lecturer at Manchester, in an intensely research-oriented group of young mathematicians, including Alan Turing (FRS 1951). Walter Ledermann was also appointed at Manchester in 1946, and then Bernhard Neumann (FRS 1959) arrived in 1948.
Graham engaged in a highly successful (and often competitive) collaboration with Bernhard Neumann and his wife Hanna Neumann, some of the results of which will be described below. As Graham himself described to his student Brian Stewart, he and the Neumanns would sometimes part on a Friday for the weekend, stuck on some part of their work, and each of them would vie to be the one who produced the breakthrough on the Monday morning.
Graham was ambitious, and began to apply for professorships at quite an early stage of his career, but Henry Whitehead encouraged him to return to Oxford rather than to seek a chair at a lower-ranked institution. In 1955 Graham was appointed as a lecturer in mathematics in Oxford, and very soon afterwards he was promoted to reader. In 1958 he was honoured with election as a Fellow of the Royal Society, and also became a senior research fellow at his former college, Balliol. Then, following Whitehead's premature death, Graham was appointed to the Waynflete professorship in mathematics, ahead of the younger Michael Atiyah (FRS 1962). This involved a move to a fellowship at Magdalen College, and he held both positions from 1960 until his retirement in 1984.
As Waynflete professor, Graham had numerous doctoral students (many of whom are listed later in this memoir). His approach to supervision was not unusual for the time, but was certainly different from current expectations. He disliked scheduling regular meetings, and preferred that his students would attend colloquia and advanced classes, not just those he conducted himself, appear for tea in the Mathematical Institute's common room on a regular basis (to engage in mathematical conversations), and come to see him only when they had something important to say or ask; and when that happened, he would put aside his own work in order to help the student, offering many ideas and explanations (usually sketched on a board), and yet he would not seek credit for those ideas, let alone eventual joint authorship.
Remarkably, on at least two occasions Graham arranged for two contemporaneous students to prove complementary halves of what became a joint theorem, including one by Sheila Oates and Martin Powell (on identical relations in finite groups), and another by Steve Smith and Peter Tyrer (on finite groups with a certain Sylow normaliser).
Graham's approach helped to develop a large degree of independence and self-reliance in most (but not all) of his students. One notable success is highlighted by an account of him telling a student: 'When I asked you a question, I didn't expect you to prove a theorem.'Sadly, a small number of students (and a few colleagues) found him intimidating, possibly because of his high academic standing and reputation, and some needed quite a lot of help, or a change of topic or supervisor. It has been reported that he once remarked that there were three types of student: those who wrote their own thesis; those for whom he wrote their thesis and they understood it; and those for whom he wrote their thesis, but they did not. Overall, he was generous with his time and help, on the basis of need.
His advanced classes covered a wide range of topics, and often contained his most recent thinking--sometimes worked out only an hour or so before the class--and occasionally he would arrive late if some detail had not quite worked out in the way that he expected. As students found in his undergraduate lectures as well, it was quite a privilege to learn from him, even if it took some time to assemble and understand the details, not all of which were given in full.
In terms of professional service, Graham co-founded the Oxford University Invariant Society in 1936 (as reported earlier), and also he served as the fifty-second president of the London Mathematical Society, from 1965 to 1967. Perhaps his greatest professional service contribution, however, was as founding editor of the Journal of Algebra, which grew to become the leading journal in algebra in a relatively short period. Graham served as its editor-in-chief from 1964 until his retirement in 1984, although his filing system was sub-optimal and so some Oxford colleagues assisted with his editorial responsibilities towards the end.
He also served as head of the Mathematical Institute at Oxford, and as chairperson of the Faculty Board, and he did well to ensure that mathematics gained an appropriate share of resources at a time when the university was expanding. Additionally, he was one of the prime movers to establish Oxford's Honours School of Mathematics and Philosophy.
Graham was renowned for his disarming ability to ask simple and seemingly innocent questions that called for evidence in support of claims made at meetings. This sometimes had the effect of exposing the ambition of certain colleagues. When new positions were to be filled, he was against attempting to make appointments in particular fields that were deemed by some to be important, but was strongly in support of appointing people doing the best mathematics.
Graham made a number of visits to the USA over the course of his career, including one for the 1960/61 academic year in Chicago, at a time when an explosion of interest in finite simple groups had occurred following some outstanding work by John Thompson (FRS 1979) that significantly extended the methods of Hall and Higman (see later). Also, immediately after he retired from Oxford, Graham spent two years as George A. Miller visiting professor at the University of Illinois, from 1984 to 1986, and then spent a year travelling around the world, visiting former students and other colleagues.
Research field Graham Higman's principal field of research was group theory, which is a branch of algebra that grew out of the study of number systems, the roots of polynomial equations and geometry, and now forms the basis of the study of symmetry.
In an abstract setting, a group consists of a collection of elements that may be combined by a single operation subject to certain laws. The given operation could be addition (of numbers or matrices or polynomials, for example), or multiplication (of non-zero numbers, non-singular matrices or non-zero rational functions), or composition (of permutations, linear/geometric transformations or symmetries of an object), or one from a range of further possibilities.
The common properties of these operations were observed and abstracted by Évariste Galois, Augustin Louis Cauchy (ForMemRS), Arthur Cayley (FRS 1852), William Burnside FRS, Ferdinand Georg Frobenius and others in the 1800s, resulting in the establishment of the theory of groups, which is now one of the cornerstones of mathematics and continues to grow and develop strongly.
Group theory plays a fundamental role in many branches of mathematics, and has important and sometimes unexpected applications in a wide range of other fields. Applications in physics, chemistry and material science are widely known, for example in the study of crystal and molecular structures. Incidentally, the structure of the Buckminsterfullerene (or C60 molecule) was already known to group theorists as a Cayley graph for the alternating group on five points decades before the molecule was discovered, and known even earlier (owing to its symmetry) by Kepler and Archimedes. More recently, group theory has played a role in the construction of efficient networks (with very good 'broadcast'properties), and 'expander graphs'(sparse networks with very good connectivity properties).
A major highlight in the development of group theory was the classification of the finite simple groups, which are building blocks for all finite groups, akin to the chemical elements. This classification took place over a period of more than 25 years, involving the collaborative effort of over 100 mathematicians from around the world, published in journal articles taking up more than 10 000 journal pages, and essentially completed by 1980.
Refinements and clarifications of the long and highly complicated proof are ongoing, but, more importantly, the classification has resulted in many further significant discoveries, and much of the work involved in it has led to the rich development of closely related fields such as representation theory and geometric group theory.
Graham Higman played a key role in the development of group theory over the course of his career, and is regarded by many as one of the three most significant British group theorists of the 20th century, alongside William Burnside and Philip Hall. He once remarked that finite group theorists were the natural successors to classical geometers, but in fact he is well known for having made original, significant and influential contributions to the study of both finite groups and infinite groups, and across a wide range of topics, as described in the next section.
Like G. H. Hardy, Graham was unashamedly a pure mathematician. (Hardy paid little attention to applications, and yet his own field of number theory lies behind the advanced systems of encryption and other forms of information security we now use on a daily basis.) He strongly held the view that pure mathematics should be studied for its beauty and its life-affirming properties, rather than its utility.
In an interview in 1987 Graham said this:
We do fundamental research, not only to acquire results solely, but because the process is an ennobling one. It is one that makes you more worthwhile than before; it is something that if you cut yourself off from, you are making yourself less human than you ought to be.This does not imply that he thought the applications of mathematics are unimportant; rather, he simply loved the 'discovery'component of pure mathematics, and for him, the applications were for others to pursue. Research achievements Graham Higman's doctoral research on units in group-rings resulted in not only a very fine thesis at Oxford but also a single-author article published in the Proceedings of the London Mathematical Society (in 1940) that helped to establish his reputation as a highly capable mathematician, and has been cited well over 100 times (1)*. In Manchester (following the Second World War), his early work covered a number of somewhat different topics, on topological spaces and linkages, and his single-author papers on those topics showed the influences of Henry Whitehead FRS and Max Newman FRS.
Arguably, his most famous piece of work took place in collaboration with Bernhard and Hanna Neumann at Manchester, on embedding theorems for groups, published in a 1949 paper in the Journal of the London Mathematical Society (2). The principal outcome of this work was a construction for what is now known as an 'HNN extension'(based on the initial letters of the surnames of the authors), defined as follows.
Let be a finite or infinite group with presentation , meaning that is a generating set for and that is a set of defining relators for (expressed in terms of the members of ), and suppose that and are isomorphic subgroups of (which could be finite or infinite) and is an isomorphism between them, then the HNN-extension of relative to α is the group .
A key observation made and proved by the three authors is that the group is embedded as a subgroup of via an injective homomorphism. An important but non-obvious consequence is that two isomorphic subgroups and of a given group are always conjugate (by some element ) inside some 'overgroup' containing as a subgroup. The latter was the motivation for this construction, which now has several important applications across a wide range of fields. For example, in algebraic topology an HNN extension enables understanding of the fundamental group of a topological space that has been 'glued'back on itself via some mapping, in the same kind of way as free products with amalgamation do for gluing two spaces along a common connected subspace. Moreover, both of these constructions are basic building blocks in the Bass-Serre theory of groups acting on trees, and HNN-extensions are now used frequently in geometric group theory and their use has been extended in the study of Lie algebras.
Shortly after the HNN paper, Graham published two more very important papers in 1951: one giving an example of a finitely-presented group that is isomorphic to a proper factor of itself (3), and another giving his famous example of a finitely-generated infinite simple group (4). These papers accelerated the growth of his high reputation.
Also while still in Manchester, Graham undertook further successful research with Bernhard Neumann and other work on unrestricted free products and topological groups, and more significantly on questions that led to a celebrated joint paper with Philip Hall in 1956, on -soluble groups and reduction theorems for the Burnside problem (7). (The latter problem, posed by William Burnside in 1902, asked whether a finitely-generated group in which every element has finite order must be a finite group. It was solved in 1964 by Evgeny Golod and Igor Šafarevič, who provided a counter-example through their solution of the class field tower problem, showing that such towers can be infinite (Golod & Šafarevič 1964).)
The Hall-Higman paper (7) contributed ideas that have influenced finite group theory ever since it appeared. In particular, this paper included a reduction theorem for the restricted Burnside problem, which asked that if m and n are positive integers, and is any finite group generated by elements and the order of every element of is a divisor of , then is the order bounded by some constant depending only on and ? Hall and Higman essentially reduced this problem to looking only at groups of prime-power exponent, and their reduction played a vital part in Efim Zel'manov's positive solution to the restricted Burnside problem in the early 1990s (Zel'manov 1990, 1991).
Another theorem in the famous Hall-Higman paper (now known as the Hall-Higman Theorem) describes the possibilities for the minimal polynomial of an element of prime-power order in a representation of a -soluble group. More generally, this paper provided the basic tools with which Walter Feit and John Thompson were later able to resolve a conjecture of Burnside by proving the Odd Order Theorem, namely that every finite group of odd order is soluble (Feit & Thompson 1963). It has also been suggested that it is no accident that a remarkable but simple idea that lies at the heart of the Hall-Higman paper is numbered Lemma 1.2.3.
Graham used HNN-extensions in another of his most famous pieces of work, now known as the 'Higman Embedding Theorem', published in the Proceedings of the Royal Society of London in 1961 (11). This theorem states that a finitely-generated group can be embedded in a finitely-presented group if and only if it is recursively presented.
As a corollary, Graham proved that there exists a universal finitely-presented group that contains every finitely-presented group as a subgroup, up to isomorphism, and indeed the finitely-generated subgroups of are precisely the finitely-generated recursively presented groups, again up to isomorphism. The Higman Embedding Theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely-presented group with an algorithmically undecidable word problem. This work was of especially wide interest since Graham needed to work at the boundaries of logic to establish the concept of relations being 'recursively enumerable'. Also part of his proof of this theorem involved an ingenious argument often referred to as 'Higman's Rope Trick', in which he almost magically made an infinite set of relations disappear. Later in his career, Graham indicated that he viewed his Embedding Theorem as his greatest achievement.
He spent the 1960/61 academic year in Chicago, at a time when an explosion of interest in finite simple groups had occurred, following the PhD thesis by John Thompson that significantly extended the Hall-Higman methodology and proved the nilpotency of the kernel of a Frobenius group. (Also the Feit-Thompson Odd Order Theorem was proved that year.) Graham realised that these developments represented the medium-term future of group theory, but he did not become fully involved in them.
Being something of a maverick, Graham embarked on a programme of characterisations of finite simple groups. These provided suitable thesis topics for his students, but neither these nor his own work formed part of the core attack on the simple group classification. He did, however, use his knowledge and expertise to devise alternative constructions for some of the sporadic finite simple groups. One was an ingenious independent construction for the Higman-Sims group (HS) as the automorphism group of a 176-point block design (13). (Note here that the 'H' in HS denotes Donald Higman, possibly a distant relative.) Another was one for Janko's third simple group , using character theory to produce a finite presentation for the group, which was verified by John McKay using a coset enumeration on a computer (14), as with another for the Held simple group (unpublished).
Graham also devised a construction for Conway's third simple group from its regular two-graph. He did not publish this, or indeed anything on the theory of regular two-graphs he developed, but he let his student Don Taylor take over his advanced class to describe and extend the latter, which has garnered recent interest among people working at the interface between combinatorics and logic.
In the later part of his career, Graham developed a strong interest in Schreier coset graphs (which he called 'coset diagrams') and their use in constructing generating pairs for alternating and symmetric groups, and also in exhibiting actions of projective linear groups. Such diagrams were a key ingredient in my own doctoral thesis and a few subsequent publications, and I have fond memories of returning to my lodgings in Oxford one evening after a trip away and from a ride in a taxi past the Mathematical Institute, seeing Graham at his blackboard working on one. Indeed the frontispiece portrait of Graham in this memoir shows a coset diagram for the full (2, 5, 5) triangle group on 66 points.
Graham also published papers on a range of other topics, including the following: varieties of groups; Lie ring methods for finite nilpotent groups; enumerating -groups; generalised polygons; the algebraic structure of groups with soluble word problem and soluble order problem; the solubility of sets of equations over groups; criteria for non-simplicity of finite groups; homogeneous relations; countably free groups; groups with relatively free verbal quotients; and infinite permutation groups. Shortly before his retirement from Oxford in 1984, he gave a course on his recent work on existentially-closed groups, which he later published jointly with Elizabeth Scott (one of his last DPhil students) in the London Mathematical Society's Monograph Series, in 1988 (16).
Influential publications At the time of completing this memoir, and according to the American Mathematical Society's MathSciNet database (https://mathscinet.ams.org/mathscinet), Graham's published work has been cited 1810 times by 1738 authors. Both numbers appear to go up by about 10 per month.
His 12 most highly cited publications have citations (as recorded on the MathSciNet database at the end of January 2022) totalling 119, 160, 88, 284, 46, 135, 92, 64, 93, 95, 125 and 110, respectively. For the non-mathematical reader, it is worth noting that these are very high numbers of citations, for many reasons. Papers in mathematics tend to cite only the previous works with content that is actually needed for the current one, so the number of citations per paper is about five on average, and also the MathSciNet database counts only citations in publications covered by the database.
Mathematical genealogy Graham Higman's academic ancestry can be traced back via his doctoral supervisor Henry Whitehead to Oswald Veblen, Eliakim Hastings Moore, Hubert Anson Newton (ForMemRS), Michel Chasles (ForMemRS), Siméon Poisson FRS, Joseph Louis Lagrange FRS and Pierre Simon Laplace FRS, and further back to other renowned mathematicians including Leonhard Euler FRS, Gottfried von Leibniz FRS and Nicolaus Copernicus.
Graham supervised at least 50 doctoral students between the early 1950s and his retirement in the mid 1980s, including those listed in a supplement to this memoir. Well over half of these former students went on to forge academic careers of their own, in various places around the world, including Jonathan Alperin, Rosemary Bailey, Donald Barnes, David Cappitt, Christopher Clapham, Michael Collins, Marston Conder, Gabrielle Dickenson (Stoy), Leslie Fletcher, Jonathan Hall, Howard Hoare, John Howie, Geoffrey Howson, Ian Hughes, Peter Lambert, Ali Liggonah, John McDermott, Thomas McDonough, Glen Mullineux, Qaiser Mushtaq, Peter Neumann, Sheila Oates (Williams), Martin Powell, Alan Prince, Christopher Rowley, Peter Rowlinson, Elizabeth Scott, Stephen Smith, Brian Stewart, Don Taylor, Sean Tobin and George Wilmers. (I offer sincere apologies to any former students whose names I have missed.)
In turn, these people supervised students who now form part of a list of well over 840 academic descendants of Graham Higman (remarkably, with over half of those being descendants of Peter Neumann). Further details can be seen at the website for the Mathematical Genealogy Project (https://www.genealogy.math.ndsu.nodak.edu).
Family life and values, and sense of humour Both Graham and his wife Ivah were active members of the Methodist Church, particularly the Wesley Memorial Church in Oxford, and Wesley's hymns had a special place in Graham's heart. While still a student, Graham became a fully accredited Local Preacher in 1936. It has been reported that Graham's Methodism was far from old-fashioned, and that he welded his faith to an understanding of evolution, and thought deeply about the role of the after-death in religion, which he described as 'one of the murkier areas of Christianity'. (Also, however, I was told on more than one occasion over 40 years ago an amusing story about Graham's being asked to preach at a Christmas service, but then on the day, forgetting that it was actually Christmas.) When Ivah died in 1981, Graham gave the funeral address himself, and he continued as a lay preacher until 2001. The minister at his memorial service (in Oxford in May 2008) remarked that Graham's faith was unconventional.
Graham and Ivah were voluntary workers with the Samaritans, as well as avid 'greens'. One of their sons, Roger, achieved national eminence in 'Friends of the Earth', and a simple cardboard coffin was used for Graham's funeral.
Also Graham was a keen bird-watcher, renowned for sometimes arriving at the Oxford Mathematical Institute in a raincoat, with binoculars around his neck, and a field guide in a small back-pack, happily reporting on his latest sighting. Indeed when he visited Brazil after his retirement, he was asked by his hosts if there was anything they could get for him, and he suggested a field guide (to carry with him when out walking), but he was then presented with two very large leather-bound volumes which contained hundreds of beautiful photos but were completely impractical.
Graham's appearance was distinctive, with an impressive beard and bushy eye-brows. He claimed that the beard resulted from a broken razor during a family summer holiday in 1966. He could look amiable to some, but scary to others.
Many of those who knew him well, however, were aware of a mischievous sense of humour. Here I will give just a few examples.
In a review he wrote in 1966 on a book on universal algebra, he wrote this:
Universal algebra is something that everyone ought to know about, though nobody should specialise in it (from which it might appear to follow that though everyone ought to read this book, nobody should have written it).He did go on to write that it would be a valuable addition to a research student's bookshelf. Occasionally the humour was even more sharp, but only when directed towards his friends. One such friend gave a lecture on 'A logical approach to HNN-extensions', after which Graham suggested to the audience that 'for a logical and intelligible account', they should read the original paper (2).
Some years later, at a short conference in Oxford he gave a lecture entitled 'The uselessness of logic', which naturally drew the attention of a large number of logicians and philosophers. In his lecture he showed that within a particular logical structure, it was possible to prove the existence of a universal object 𝒰 that would have every countable group as a homomorphic image, provided that 𝒰 had more than one element, but that within that logical structure, it was impossible to prove the latter requirement. Of course he had to field a large number of criticisms and questions at the end, but he was ready for all of them, with his enigmatic calm demeanour.
Here I will add one more thing that I witnessed first hand, not so much about his mischievous sense of humour but about an incident that tested his good humour. The weekly Algebra Seminar was held from 5pm to 6pm, after which the speaker and other participants would traditionally walk across St Giles for a drink and further discussion about the seminar topic at the 'Eagle and Child'(a favourite haunt of C. S. Lewis and J. R. R. Tolkien in their day), which opened at 6pm. One time, however, the visiting speaker finished very early and there were almost no questions, so the seminar ended over 20 minutes early. Poor Graham decided to walk home, somewhat annoyed, not only because advantage had not been taken of the full hour, but also because he did not want to be seen to be waiting around solely for the pub to open.
Closing remarks
Above all, aside from being an outstanding mathematician, Graham Higman was a proudly upright and yet humble individual.
His generosity of spirit is evident in his work for his church and the Samaritans, the stand he took against war, the love for his wife Ivah and their family, his concern for his students and his sense of humour. In a letter he sent to the Royal Society after being asked about aspects of this memoir (well in advance of it being written), he stated that he was uninterested in the personal aspects of his biography--but would be quite happy if his mathematical contributions were properly assessed--and also that he did not like any photograph of himself that he had seen, but he really liked his portrait by Norman Blamey and asked that it could be included. (Incidentally, one of the people who commissioned that portrait exclaimed when first shown the painting, 'That's the Graham we all know and love', imagining that Graham could be thinking 'I've proved a good theorem, now you lot do better!')
Graham's physical health deteriorated in his later years, but his mind did not. He spent the last few years of his life in a nursing home, but in 2007 he made a final (and well-dressed) public appearance at a conference in Oxford marking his ninetieth birthday (and the fiftieth anniversary of the Hall-Higman paper), much to everyone's surprise and pleasure.
It seems appropriate to conclude with this comment made about Graham by Don Manley in The Independent shortly after Graham's death (at the age of 91) in Oxford on 8 April 2008: 'In an age when successive governments want science and mathematics to justify themselves in terms of the national economy, this lovable and grumpy old man seemed to stand for something else that mattered much more.'
Awards and recognition
In addition to honorary degrees, Graham Higman was accorded the following honours for his outstanding work in mathematics:
1951 Morning Speaker, British Mathematical Colloquium
1957 Plenary Speaker, British Mathematical Colloquium
1958 Fellow of the Royal Society of London
1960-1984 Waynflete Professor of Mathematics, Magdalen College, Oxford
1962 Berwick Prize, London Mathematical Society
1965-1967 President of the London Mathematical Society
1974 De Morgan Medal, London Mathematical Society
1979 Sylvester Medal, Royal Society of London
Two others were the commissioning of the Norman Blamey portrait of Graham (shown as the frontispiece portrait of this memoir), and the establishment of the Higman Room in Oxford's Mathematical Institute to mark his retirement.
Acknowledgements
An invitation to write this memoir was initially made to Dr Peter Neumann (University of Oxford), one of Graham Higman's first doctoral students, but sadly Peter suffered a stroke and found he was unable to complete the task, and shortly afterwards died from Covid-19, in December 2020. I felt honoured to be invited to write the memoir in Peter's place, and I hope it comes close to what he would have written.
Next, I wish to acknowledge a variety of sources of information: the Bodleian Libraries online search facility (https://www.bodleian.ox.ac.uk/collections-and-resources/solo); the MacTutor History of Mathematics Archive (https://mathshistory.st-andrews.ac.uk); and the obituary articles that appeared in The Independent and The Telegraph in May 2008 and the Oxford Mathematical Institute Newsletter in the spring of 2009 (Anon. 2008; Collins 2008; Anon. 2009). The frontispiece portrait of Graham is reproduced with the kind permission of the Mathematical Institute, Oxford. It was painted by Norman Blamey in 1984, commissioned by former students and friends (who gave it to the institute) and it now hangs on a wall on the ground floor of the institute in the north wing of the Andrew Wiles Building at Oxford. I would also like to thank Stephen Blamey (son of the artist), Mike Giles (the current head of the Oxford Mathematical Institute) and Brian Stewart and Michael Collins (two of the portrait's commissioners) for arranging and endorsing permission to use it.
Last, but by no means least, I wish to heartily thank Rosemary Bailey, Martin Bridson, Peter Cameron, Gareth Jones and Brian Stewart for suggestions and additional points that greatly helped to improve this memoir.