Memories of Peter Neumann

What follows are memories, and at my age my memory is not totally reliable, so don’t take any of this as absolute truth.

But it is important to say that Peter was one of the kindest people. I owe him a huge amount, but he was always supportive and never once reminded me of this debt. However, with his kindness, he was in no way a soft touch.

Peter was born in Hull, and always regarded himself as a Yorkshireman. He told me that when he was born his mother was living in a caravan, writing her thesis on a portable typewriter; to get him out from under her feet, he was put on a lead and allowed to crawl round the garden.

Sometime around 1970, the British Mathematical Colloquium was in York, and Peter took me and another of his students, Graham Atkinson. He rented a car and we drove up to York. Along the way, after a stop, he suggested that Graham should drive for a bit. It was a very windy day, and Graham lost control of the car on the motorway; we crossed the central reservation and ended up on the hard shoulder on the wrong side, completely unscathed. Peter took over and drove the rest of the way. But he regarded it as important to get Graham back behind the wheel of the car, so he would not lose confidence; so at the meeting they went out driving a couple of times.

Back then, contributed talks at the BMC were in subject-specific “splinter groups”, to which you signed up at the meeting. John Conway was there; it may have been his first BMC as well as mine. He thought that if he signed up for several splinter groups, he would increase his chances of being offered a place in one of them. Inevitably he gave all the talks he signed up for. I remember that one of them was about octonions. He showed us an identity which he called the “plaiting rule”; but his Liverpool accent was sufficiently thick that I heard it as the “punting rule”. I couldn’t see what it had to do with punting, but I was aware that punting at Cambridge was rather different from punting at Oxford …

Peter was only six years older than I, but I was his sixth student. When I came to Oxford, the intention was that I would be Graham Higman’s student. But Graham was on leave at the time, so I began with Peter, and transferred to Graham when he got back. After a short time he decided to trasnsfer me back to Peter. (It is claimed that he said, “You ask too many questions; you can go back to Peter”, though I don’t actually remember him saying this.)

Peter was the ideal supervisor for me. He gave me Wielandt’s book on permutation groups to read at the start. After that, he let me find my own direction, but he was always there with suggestions and questions. At the time, he had just extended Wielandt’s theorem on primitive permutation groups of degree twice a prime to degree three times a prime. The argument was based on Donald Higman’s theory of coherent configurations, so I had an early acquaintance with that. One of the rare examples of such a permutation group is PSL(2,19), with degree 57. Fortuitously it happens that one of the orbital graphs is 2-arc-transitive, though we didn’t call it that back then: the point stabiliser is A₅, acting 2-transitively on an orbit of size 6. (This graph is now called the Perkel graph, after Manley Perkel, who studied it in the late 1970s; but perhaps Peter Neumann should get some credit.) W. A. Manning had shown that if a primitive but not 2-transitive group has the property that the point stabiliser is 2-transitive on an orbit of size greater than 2, then this is not the largest orbit. I happened to notice that the reason for this is that the group is transitive on paths of length 2 in the orbital graph, and so points at distance 2 from α form an orbit of the stabiliser of α, larger than the starting orbit. This short proof of a theorem that cost Manning much more effort was the subject of my first paper, and led to my thesis topic.

Peter had four students in my year; as well as me, there were Gareth Jones, Elizabeth Morgan (now Billington), and Mary Tyrer. Gareth and Mary got married and spent their career in Southampton. My relationship with Liz was a bit different: we changed places. I moved from Brisbane to Oxford; she moved in the other direction, and finished her thesis with Sheila Oates. We remained on opposite sides of the world, though we managed to continue as friends. So Liz and I are mathematical cousins rather than siblings.

Peter ran a “Kinderseminar” for his research students and selected other people. When I was appointed to a fellowship at Merton College, and had students of my own, I tried to replicate this on a small scale; but, to my great delight, soon Peter invited me and my students to join the Kinderseminar. We would meet at 11, have coffee and chat, and then someone would talk about some interesting mathematics. At the end of the talk, Peter would discretely take the student to one side and go over the talk, not criticising but making friendly suggestions about what could have been done better. If you want to know why all of Peter’s students are good lecturers, look no further than this.

Peter was one of the pioneers of the combinatorial approach to permutation groups associated with the names of Donald Higman and Charles Sims, and I was very fortunate to learn about this early on. But he did much more. He has an influential paper with both his parents and Gilbert Baumslag on varieties generated by a finitely generated group, published in 1964 when he was in his early 20s. His doctoral thesis contained results about wreath products, among other things. He was perversely proud of the fact that it was the only doctoral thesis ever to be stolen from the Whitehead Library of the Oxford Mathematical Institute!

Peter was always interested in the history of algebra. I recall one occasion when I was in his rooms and we were discussing something, I no longer remember what, but probably connected with the O’Nan–Scott Theorem. Peter reached up to a high shelf and pulled down Camille Jordan’s Traité des Substitutions; he turned to the page where Jordan proves most of the O’Nan–Scott Theorem. In the case where the socle of a primitive group is non-abelian and not simple, Jordan clearly states the crucial case division that leads to wreath products on one hand and diagonal groups on the other. I feel that a fitting tribute to Peter would be on the history of the O’Nan–Scott Theorem: What exactly did Jordan prove? Why was it forgotten? What happened subsequently?

All of 19th century algebra was Peter’s playground, and in particular he became an expert on Galois, editing all of his writings with detailed commentary.

He gave a lot of time to serving various organisations in various capacities, especially the London Mathematical Society, the British Society for History of Mathematics, and the UK Mathematics Trust. With the LMS, he dealt with publications for some time. This was back in the days of hot-metal printing. The LMS printers were a small specialist firm in Leytonstone, who may have been called Hodgson and Son; Peter used to make fairly frequent trips to East London to discuss printing matters with them. (All his students will testify that he instilled in them a sense of how printed mathematics should look, with simple but oft-forgotten rules such as: Don’t start a sentence with notation; don’t put two formulae adjacent in a sentence.)

It must have been at about this time that the LMS started up its Bulletin, to contain short papers, as well as surveys, records of meetings, book reviews, and obituaries. I don’t know what role Peter had in establishing this journal. But I was fortunate to have my first paper published in the first volume, along with John Conway’s paper introducing his groups.

He was awarded an OBE in the New Year Honours in 2008, for his services to education; I think this primarily referred to his work with the UK Mathematics Trust.

Peter had a hand in getting me involved in the connections between permutation groups and transformation semigroups, especially synchronization. The push came from the coincidence of three things. First, Cristy Kazanidis, who had been a postdoc at Queen Mary, was back in London and came to see me looking for a research problem. I suggested that we could look at cores of rank 3 graphs. So when I needed it, I had the information about graph homomorphisms required to characterise synchronizing groups to hand. Second, Robert Bailey, my former student, came back to Britain for Christmas, and told me about a conversation he had had with Ben Steinberg about synchronization at a bus stop in Ottawa. This sounded interesting to Robert, but halfway through, Ben’s bus had come along and the conversation was never finished. Finally, someone asked me for the notes of an Oberwolfach meeting on permutation groups the previous summer. On looking them out, I found also the notes of Peter Neumann’s talk on synchronization, based on an exchange he had had with João Araújo. This got me in touch with João, and the rest is history (or may be one day, it is still going on!). João told me later that he had had the big idea of using advances in group theory based on CFSG to make progress on semigroup theory. When he talked to Peter, he had just had a paper proposing this idea rejected (see the preceding post). But Peter reassured him that it was beautiful mathematics and encouraged him to continue.

One final memory. One year in the 1970s, there was a finite geometry conference at the Isle of Thorns (the University of Sussex conference centre in Ashdown Forest). Peter, Jan Saxl and I decided to walk there from Oxford. We took three and a half days, staying overnight in Reading, Guildford, and East Grinstead, and arriving at the conference about lunchtime. It was beautiful autumn weather. Peter and Jan each had to skip a bit of the way because of minor injuries, and use public transport. Now, after this dreadful year, I am the only survivor of that walk …