I learned much from Roger and this book; what a huge and exciting subject number theory is. [A number theory] problem took over my mind and I have always believed that it was the main reason that I failed one of my A-level GCE mathematics papers in June 1958, failed to win a state scholarship, and had to repeat the examinations in 1959. It may, however, have contributed something towards my winning a Hastings Scholarship to The Queen's College, Oxford in the Autumn of 1958.

Mathematics was always part of the Neumann household, with students from the university being welcomed to evenings of chat, puzzles and fun. In summer, it was not unusual for the full family of seven to spend a fortnight cycling to and from Scotland or elsewhere in the UK so that Bernhard and Hanna could attend conferences. One such was the Mathematical Colloquium organised by the Edinburgh Mathematical Society at the University of St Andrews every four years. As the children grew older, they, too, would attend selected sessions. Music was also part of family life, with several of the children playing different instruments, so that chamber music could be played, or recorder ensembles with descant, treble and even a bass recorder.

For several years the family was split, with Bernhard working in Manchester University while Hanna kept the family going, as well as working at the University of Hull. Bernhard would come back for alternate weekends and the vacations, sometimes by train but often by bike. For Peter's final year at Hymers College, Hanna had also managed to find work in Manchester so the family left Hull but, to finish his schooling, Peter lodged with another mathematical family, John and Beryl Shepperd and their children.

This paper studies interrelations of varieties of groups, that is of classes of groups defined by laws (also known as identical relations) or, in yet different terminology, of equationally defined classes of groups. The set of varieties of groups carries a natural lattice order relation, by inclusion, and it is given an algebraic structure by the definition of certain operations: the lattice operations that stem from the order relation, and multiplication and commutation.

A number of properties of this algebraic structure were derived in [Hanna Neumann, 'On varieties of groups and their associate near-rings' [1956]] thus it is known to be multiplicatively an ordered semigroup with unit element and zero, with cancellation of equal non-zero right-hand factors from inclusions and equations, and with multiplication on the right distributing over the lattice operations and over commutation. Every non-one and non-zero variety is, moreover, expressible as a product of indecomposable varieties, that is of varieties that are not products except trivially.

Among the problems that are posed but left open in [Hanna Neumann's paper] the most difficult is that of the uniqueness or otherwise of the factorisation of a variety into indecomposable varieties; there is also the problem whether multiplication on the left behaves like multiplication on the right: whether equal non-zero left factors can be cancelled from inclusions or from equations, and whether multiplication on the left distributes over the other operations. All these questions are answered in the present paper ...

... the College's reaction was sternly to deprive him of his scholarship, but then to make him an ex gratia payment, of a sum exactly equivalent to a scholarship.

I married the year I left Somerville and taught for a year at Milham Ford School while my husband ... was still an undergraduate. Then our family started and my career went on hold. ... we have brought up three children: David, born in 1964; Jenny, born in 1965; and James, born in 1968.

This thesis records an attempt to prove the two conjecture:
Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points.
Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most $\large\frac{1}{2}\normalsize m$. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are:
Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than $\large\frac{1}{4}\normalsize (n+3)$ points.
Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points.
Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than $\large\frac{1}{2}\normalsize m$.
Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than $\large\frac{7}{18}\normalsize m$.
Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

I owe my interest in the problems described in this thesis to a rich correspondence with Dr James Wiegold. ... I am grateful to my supervisor, Professor G Higman, F.R.S., for his kind encouragement and for many suggestions, and for a wide background of valuable instruction and supervision more generally; and to Dr Wiegold who, unwittingly at the time, set me off on this investigation, and whose friendly influence has helped in many, many ways.

In have taught all aspects of pure mathematics to first and second-year undergraduates; taught algebra, logic, number theory, combinatorics, history of mathematics to advanced undergraduates; and taught algebra, history of algebra to postgraduate students. Thirty-six students have earned an Oxford DPhil under my supervision (35 in algebra, 1 in history of mathematics).

I have undertaken research in algebra - mainly group theory - and its history. Broad areas to which I have contributed are: finite permutation groups; infinite permutation groups; computational group theory; varieties of groups; infinite soluble groups; statistical questions of group theory; group enumeration; applications of group theory in combinatorics; Nineteenth Century history of group theory. I have published approximately 75 research articles in mathematics (about half of them written jointly with colleagues), 8 research articles in history of mathematics, a textbook of group theory (joint with G A Stoy and E C Thompson), lecture notes on permutation group theory and on computational algebra, an edition (joint with A J S Mann and Julia Tompson) of the papers of William Burnside, and a number of reviews and minor expository items.

I have learnt and taught mathematics at all levels from school to postgraduate; the art of lecturing and university teacher training; public understanding of mathematics; relationship of mathematics with other disciplines - especially computation, and philosophy.

Peter himself should have written much more - but he always said "There are already too many books". While this is true, more books by Peter would have benefitted us all.

Let $\Omega$ be a finite set of size n and G the subgroup of $Sym (\Omega )$ generated by a collection $g_{1}, g_{2}, ..., g_{k}$ of explicitly known permutations. How do we compute the composition factors of G?

This is one of the many mathematical questions where theory and practice differ greatly. In theory we might work through the elements of G one by one seeking g such that the normal closure $N := < g^{G} >$ is a non-trivial proper subgroup of G; if no such g exists the G is simple, otherwise we treat N and G/N in the same way until all the composition factors have been found. In practice this does not work. There are two reasons. The first is obvious - such a search will be absurdly slow even for permutation groups of quite small degree. The second is more subtle and more significant - we have at present no satisfactory way of handling a quotient groups G/N as a permutation group except in certain special cases.

Many of the ideas in this paper were hatched during the London Mathematical Society Symposium on Computational Group Theory held in Durham from 30 July to 9 August 1982, and during a visit that I paid to the University of Sydney from 20 March to 20 April 1983. I am grateful to the organisers of that conference for inviting me to come there - as a stranger in paradise - and to the Mathematical department of the University of Sydney for its genial hospitality.

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. ... 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.

The UKMT was established in 1996, and Peter was the first Chairman of its Council, a position that he held until 2004. Peter was a very hands-on Chairman of UKMT. He was whole-heartedly involved with many of the Trust's activities, such as mentoring programmes for able students, and marking-weekends, in some of which Sylvia also joined. During the early years of the UKMT people from different backgrounds worked together to build the new organisation with little administrative support. Additional stresses arose when the new UKMT agreed at very short notice to take on the planning, funding and organisation of the International Mathematical Olympiad (IMO) for 2002, after the original host country had to withdraw. This posed a not inconsiderable risk to the future viability of the UKMT. It was thanks to Peter's commitment to enhancing the mathematical experience of talented young people, his acute mind, and his great charm and courtesy that the difficulties were overcome.

... in recognition of his contribution to and influence on research into diverse branches of group theory, and for his broad-ranging service to British mathematics over many years.

For services to Education.

I owe so much to Peter - and so do many many others. His contribution to mathematics goes far beyond his own mathematical discoveries, significant though these are. He taught, inspired and encouraged so many others, and was much loved as well as much admired.

During his long professional life he enriched the lives of many people, not only mathematically but through his consideration and kindness. Many will also have enjoyed social occasions at his home and in college where he was a most generous host. He leaves a legacy of which his family can be immensely proud.