I first became interested in mathematics in general when I was in school. In the same way as most school children choose a subject, it was the subject I was best at.

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."

When I was an undergraduate I was interested in several branches of mathematics. When I became a research student under Henry Whitehead, who was of course a topologist, he put me on to a problem in group theory with the intention that it should be applied later on in topology. I got as far as the group theory but I got stuck there so in a way you could say that I am a failed topologist and that's why I'm a group theorist.

I should like to thank Mr Whitehead for many helpful suggestions and much good advice that he has given me, both in the investigation of these problems and in the writing of this paper.

It is trivial that a group all of whose elements except the identity have order two is Abelian; and F Levi and B L van der Waerden have shown that a group all of whose elements except the identity have order three has class less than or equal to three. On the other hand, R Baer has shown that if the fact that all the elements of a group have orders dividing n implies a limitation on the class of the group, then n is a prime. The object of the present note is to extend this result by showing that if M is a fixed integer there are at most a finite number of prime powers n other than primes, such that the fact that all the elements of a group have orders dividing n implies a limitation on the class of its Mth derived group.

Any finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented.

Well, in the first place, like many other research mathematicians, I do what I do, not only because I think it is important but also because I think it's interesting. The problems one has to deal with in group theory are challenging, they are intellectually difficult. When one has got into a problem it's not so much the question of how much it has an application to the real world that concerns you, it's a question of exercising your mind, exercising the control the mind has over the things it understands. Of course it is true that group theory has applications in the sciences. Perhaps we should say for the benefit of most of the people who might be listening to us and have no idea what group theory is, that group theory is in fact the study of symmetry. What a group theorist is concerned with is the possible amounts of symmetry that an object can have, symmetry in the abstract is our study and because everything has some degree of symmetry and because a physical scientist for instance who is studying a problem finds it easier to deal with it if he takes symmetry into account he needs to know what kinds of symmetries there are and therefore needs to know the results of group theory. It is in that way that group theory acts as a servant to other sciences as well as being an interesting subject in its own right.

As Waynflete professor, Graham had numerous doctoral students. 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. ... Graham's approach helped to develop a large degree of independence and self-reliance in most (but not all) of his students. ... 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.

Relaxation came from bird-watching, and Higman would often arrive in the morning at the Mathematical Institute in a shabby raincoat, binoculars around his neck, to report on the latest sighting. He remained active in the Wesley Memorial Church in Oxford with a particular love for Wesley's hymns, and he participated in one of its book reading groups until about three years ago. For many years, too, he was involved with the Samaritans.

This volume grew out of lectures given by Higman at Oxford in 1983 and 1984 as recorded and amended by Scott. It is not a comprehensive work on existentially closed groups but rather contains an ample selection of topics written at an easily accessible graduate level. Both algebraic and model-theoretic aspects of existentially closed groups are highlighted.

The subject of existentially closed groups currently occupies a special place in the model theory and logic of algebraic systems. This is due to a number of convergent factors, not the least of which is that this subject performs a type of wedding between recursion theory (the foundation of mathematical logic) on the one hand and group theory (the foundation of the analysis of mathematical structure) on the other. In another sense, it and related topics in other systems provide a testing ground for the classificatory concepts of model theory. Existentially closed groups provide an exemplary case in which powerful techniques are available to explore the details and potentials of logical complexity in a class of systems well known for their intrinsic beauty.

This paper can, I think, best be considered as a sort of five finger exercise. In it we polish the techniques from a corner of group theory by using them to investigate the structure of a particular infinite permutation group. However, the group that we consider though particular is a fairly natural one, and while the results have a familiar general shape they include one detail which to me was quite unexpected. These facts perhaps constitute a challenge to place the results in a more general context; if so I make no attempt to meet it.

Participants felt themselves privileged when Graham Higman decided only the day before that he would attend both for the opening and for some of the lectures.

Roger joined the Network of Wellbeing team in late 2014 to take on a leading role in developing our strategy. Roger has over 25 years' experience of campaigning mainly at Friends of the Earth, including locally, nationally and internationally on issues as diverse as climate change, biodiversity, resource use and economics. Roger is also a Director of the Flameworks Creative Arts Facility in Plymouth, where he lives with his partner, Karen and her three daughters. He is at his happiest exploring the countryside with Karen or working his allotment with his brother, Joe.