# Donald Gordon Higman

### Quick Info

Vancouver, British Columbia, Canada

Ann Arbor, Michigan, USA

**Donald Higman**was an Canadian mathematician best known for his work in group theory and in particular for the discovery of one of the sporadic simple groups.

### Biography

**Donald Higman**'s parents were Leslie Higman and Hazel Delia Barrett. Leslie Higman was born in St Columb, Cornwall, England on 24 December 1899. At the time of the 1901 census he was living in Newquay, Cornwall, England, but in 1906 the family emigrated to Canada. In 1921 he was living in Vancouver South, British Columbia, Canada and on 23 January 1926 he married Hazel Delia Barrett in East Vancouver, British Columbia. Hazel Barrett was born in Seattle, Washington, USA, in March 1905. The family were still living in Seattle when her sister Evelyn Thelma Barrett was born in 1906 and when her brother Andrew John Barrett was born in 1911 but by the time her sister Elizabeth L Barrett was born in 1914 the family had moved to Canada and were living in British Columbia. Donald had one younger sister, Leslie Joan Higman, born in 1929.

Higman was educated in Vancouver where, after graduating from High School, he entered the University of British Columbia. On 14 August 1948 he married Kathleen Elizabeth Scott (1926-2012), always known as Betty, in Vancouver. She had been born on 23 August 1926 in Caren, Saskatoon, Saskatchewan. The Higmans had five children.

Cecilia Krieger had been on the faculty at the University of Toronto since 1928 and by 1942 she was an assistant professor there. She spent 1948 as a visiting lecturer at the University of British Columbia and during that year she lectured to Higman. Realising that he had great potential as a mathematician, she urged him to apply to the University of Illinois at Urbana-Champaign so that he might continue studying mathematics at the graduate school. She almost certainly made this suggestion knowing that Higman was enthusiastic about algebra and that Reinhold Baer held a chair there. Higman received his B.A. degree in 1949 from the University of British Columbia and, having been accepted for the University of Illinois graduate school, began his studies there advised by Baer. He received his M.A. in 1951 and in the following year his Ph.D. for his thesis

*Focal Series in Finite Groups.*

While at the University of Illinois Higman made a number of important contacts, the two most important being Michio Suzuki and Hans Zassenhaus. Suzuki had started research in the University of Tokyo's graduate school in April 1948 but in January 1952 he arrived at the University of Illinois having been awarded a graduate fellowship. Suzuki wrote in one of his articles (see [1]):-

When I [Suzuki] went to Illinois in 1952, Donald Higman was there and had just completed his Ph.D. thesis on focal subgroups. Partly because I knew the name of Higman already, by his paper on homomorphism correspondence of subgroup lattices, soon we became very good friends.Hans Zassenhaus had accepted a chair at McGill University, Montreal, in 1949 and Higman met him at a Canadian summer mathematics program in either 1951 or 1952. After Higman graduated with his Ph.D., he was able to spend the two years 1952-54 at McGill University funded by a National Research Council Fellowship. His first publication

*Lattice homomorphisms induced by group homomorphisms*appeared in 1951 while he was still a research student and four further publications appeared during the two years at McGill University, namely:

*Focal series in finite groups*(1953);

*Remarks on splitting extensions*(1954);

*Modules with a group of operators*(1954); and

*Indecomposable representations at characteristic p*(1954). The first of these four papers is, of course, a paper containing the main results of his thesis about the focal subgroup theorem [1]:-

Don Higman's focal subgroup theorem is an insightful result about intersections of normal subgroups of a finite group with a Sylow p-subgroup. ... Higman's theory of the focal subgroup of a Sylow subgroup was a basic tool in local analysis in finite group theory. It could be viewed as a contribution along the lines of Burnside, Frobenius, Grün, et al. to the determination of quotients that are p-groups (p is a prime number). It turns out that the focal subgroup theory is logically at the centre of p-local group theory. ... In the focal subgroup paper, there is a final remark about a communication with Brauer who reports overlaps with results from his paper on the characterization of characters [A characterization of the characters of groups of finite order (1953)]. Higman states that those results follow from the focal subgroup theory. ... Richard Brauer essentially agreed with Don's assertions. ... Brauer was impressed with Higman, and this may have influenced Don's success in getting a job at the University of Michigan (for Brauer had been a member of the University of Michigan faculty during 1948-1951 before moving to Harvard).The second of the above mentioned four papers,

*Modules with a group of operators*(1954), contains the important definition of a relatively projective (or injective) module for the group algebra of a finite group and proves the famous criterion for relative projectivity that is today named for him, being known as 'Higman's criterion'.

The references [1], [3], [4] and [5] all state that Higman spent the two years 1954-56 on the faculty of Montana State University. This university is situated in Bozeman, Montana, yet in 1955 Higman and his wife were living in Missoula, Montana, which is the city where the University of Montana is based. In 1956 he was appointed as a visiting assistant professor of mathematics at the University of Michigan where he spent the rest of his career. He was appointed assistant professor of mathematics in 1957, promoted to associate professor in 1960 and to professor of mathematics in the College of Literature, Science, and the Arts in 1963. At the University of Michigan he became a colleague of Jack E McLaughlin and the two worked closely together writing three joint papers

*Finiteness of class numbers of representations of algebras over function fields*(1959),

*Geometric ABA-groups*(1961), and

*Rank 3 subgroups of finite symplectic and unitary groups*(1965). For many years the two discussed a wide range of different aspects of mathematics and each gained much from these discussions. Their common interests included general algebra, group theory, representation theory, and cohomology.

Higman played a large role in creating an extremely friendly atmosphere in the Mathematics department at Michigan, where there were many parties and constant forms of hospitality. The Higman family, Donald, Betty and their children, hosted many parties in their home [1]:-

Betty worked at the Library for the Blind and Physically Handicapped. Don was an active member of the Flounders and the Ann Arbor Track Club, and he frequently rode his bicycle. Don played water polo well into his later years. Bob Griess remembers many times being at home, noticing Don jogging past as he expelled his breath in loud bursts.In 1967 Higman made a major breakthrough when, working with Charles Sims, he discovered the previously unknown sporadic simple group now known as the Higman-Sims group. Let us note here that Charles Sims had been an undergraduate at the University of Michigan and had been taught algebra by Higman. Both Higman and Sims were at the conference 'Computational Problems in Abstract Algebra' held in Oxford in that year. At the conference Marshall Hall lectured on the construction of the Hall-Janko sporadic simple group as a rank-3 permutation group on 100 points and this prompted a discussion. Before, during and after the conference dinner held on Saturday, 2 September 1967, the two mathematicians developed their ideas. Using Higman's theory of rank-3 groups, they constructed the Higman-Sims group as a primitive permutation group of degree 100, with order 44,352,000 and rank 3. This group is a rank 3 extension of the Mathieu group $M_{22}$. Higman and Sims described the group in the paper

*A simple group of order 44,352,000*which appeared in

*Mathematische Zeitschrift*in 1968.

Peter Cameron writes [2]:-

... the Higman-Sims group, was found without any recourse to computation at all. It is a subgroup of index 2 in the automorphism group of a graph on 100 vertices, constructed from the 22-point Witt design. The graph had been constructed earlier by Dale Mesner ... Indeed, from my point of view, Higman and Sims were the two people who introduced graph theory into the study of permutation groups. I was lucky enough to be in on the ground floor, beginning my doctoral studies in 1968 (the year after the Higman-Sims group was found, though I wasn't yet in Oxford on that memorable occasion).Peter Cameron writes in [1]:-

It was very important to me that Don Higman spent part of the year 1970/71 in Oxford, for several reasons. First, he gave a course of lectures. I was one of two students (the other was Susannah Howard) responsible for producing printed notes after each week's lectures and then compiling them into a volume of lecture notes published by the Mathematical Institute, Oxford, under the title 'Combinatorial Considerations about Permutation Groups'. These notes contained a number of ideas that appeared only later in conventional publications; I was in the very fortunate position of having a preview of the development of the theory of coherent configurations. ... Fourth, and most important for me, Don arranged for me to spend a semester at the University of Michigan as a visiting assistant professor. This was a very productive time for me. I was allowed an hour a week in which I could lecture on anything I liked, to anyone who wanted to come along. I did a vast amount of mathematics there, and made many good friends.Michael Aschbacher writes [1]:-

I probably first met Donald Higman at a meeting in Gainesville, Florida, in 1972, organized by Ernie Shult. However, my strongest early memories of Higman are from a two-week meeting in Japan in 1974 organized by Michio Suzuki. This was only my second trip abroad, and one of my favourites. A number of wives also took part, including my wife Pam. Both Donald and Graham Higman attended the meeting, and I recall Pam referring to them as "the Higmen". As I recall the two were connected in our minds, not just because of their names, but because both had extremely luxuriant eyebrows. ...At the University of Michigan, Higman served on many departmental committees including the executive committee, the doctoral committee, and the master's committee. He also served on the University's grievance committee. He supervised fifteen doctoral theses and several of his academic children went on the gain academic posts resulting in around 25 academic grandchildren, a number which, of course, will continue to increase.

Over the years Higman made many visits abroad to conferences and other events. For example he spent sabbatical and academic leaves in Eindhoven and Giessen, was a visiting professor at Frankfurt, a visiting senior scientist at Birmingham and Oxford, and a visiting fellow at the Institute for Advanced Study in Canberra, Australia. He organised meetings in Oberwolfach, Germany, the first being "Die Geometrie der Gruppen und die Gruppen der Geometrie under besondere Berücksichtigung endliche Strukturen" Ⓣ held 18-23 May 1964. In June 1966 he was in Japan and, in that month, gave the lecture 'Remarks on Finite Permutation Groups' at the University of Tokyo in which he described the discovery of Janko's simple group and discussed Donald Livingstone's construction of it. He attended the summer Institute of Mathematics at Patna University, India, in 1966 and a conference in Patna, Bihar, India in September of that year. He visited Oxford for a few months in 1971 and gave a course of lectures entitled 'Combinatorial Considerations about Permutation Groups'. In 1974 he attended the two-week Sapporo Conference in Japan organised by Michio Suzuki. He enjoyed a sailing holiday in Friesland, The Netherlands, in June 1978 and attended the American Mathematical Society Summer Research Institute on Finite Group Theory at the University of California, Santa Cruz, 25 June - 20 July 1979. He attended the Research Institute for Mathematical Sciences at Kyoto University, Japan, 21-24 November 1994.

Among the many honours given to Higman, we note especially his invitation to lecture at the International Congress of Mathematicians in Nice in 1970 when he lectured on his theory of rank-3 groups. In 1975 he was awarded the Alexander von Humboldt Stiftung Prize.

He retired from active duty on the faculty of the University of Michigan on 31 May 1998. The Memoir [5] was produced when he was made professor emeritus. It states, "The Regents now salute this faculty member by naming Donald G Higman professor emeritus of mathematics."

Robert Griess writes ([3] and [4]):-

After retiring, the Higmans designed and built a home on the shore of Grand Traverse Bay in the northwest corner of Michigan's Lower Peninsula, where they enjoyed spending summers. Professor Higman had cancer and volunteered to be part of an experimental research study at the University of Michigan Comprehensive Cancer Center, contributing in whatever way he could to find a cure for the disease. It was important to him to be a contributing member to society even as he went through his own personal struggle with the disease.In February 2009 the

*Michigan Mathematics Journal*produced a special memorial issue for Donald Higman.

### References (show)

- E Bannai, R L Griess, Jr, C E Praeger and L Scott, The Mathematics of Donald Gordon Higman,
*Michigan Math. J.***58**(2009), 1-28. - P Cameron, Charles Sims,
*Peter Cameron's Blog*(25 October 2017). https://cameroncounts.wordpress.com/2017/10/25/charles-sims/ - R L Griess, Memorial. Donald G Higman,
*University of Michigan*. https://www.lib.umich.edu/faculty-history/faculty/donald-g-higman/memorial - R L Griess, Obituary. Donald G Higman,
*University of Michigan*. https://www.lib.umich.edu/faculty-history/faculty/donald-g-higman/obituary - Memoir. Donald G Higman,
*University of Michigan*. https://www.lib.umich.edu/faculty-history/faculty/donald-g-higman/memoir

### Additional Resources (show)

Other websites about Donald Higman:

### Honours (show)

Honours awarded to Donald Higman

Written by J J O'Connor and E F Robertson

Last Update September 2018

Last Update September 2018