**Brian Hartley**'s parents were Thomas Hartley (1907-1969) and his wife Annie Kay (1912-1987). Thomas Hartley, a joiner, married Annie, a cotton mill worker, in 1934. They had two children, Brian (the subject of this biography) born in 1939 and a daughter Mavis born three years later. Thomas and Annie Hartley lived with Annie's mother Amelia Kay for the first twelve year of their marriage and both continued working. This meant that Brian was mostly brought up by his maternal grandmother for the first seven years of his life.

In the autumn of 1943 Brian began his schooling at Spring Hill Infants School, Accrington. Brian didn't spend all his primary education at this school for he attended the Peel Park Infants in Accrington in 1945-46. In the summer of 1946 the family moved into a small terraced house in Fife Street, Accrington and shortly after, Brian attended Spring Hill County School. Spring Hill school was built in 1899 but the infant school was only built in 1932. Brian's performance at school was outstanding and, in 1950, he won a place at Accrington Grammar School. The famous mathematician Harold Davenport had been educated at this school thirty years before Hartley began his studies there [3]:-

Hartley loved sports, particularly cycling and running. I [EFR] knew Brian quite well and he told me about his love for both cycling and running. There were different aspects to liking these sports for Brian. He was a natural sportsman taking delight in exercise but he also loved the outdoors and the countryside, particularly the hills. There was yet another aspect that Brian loved which was his natural competitive spirit so that when he was at school he enjoyed both the training and the competition that came from being a member of the cross-country team. He was also a member of the local Cycling Club when he was at school in Accrington. When I knew Brian after he became a lecturer he often came into the department on a Monday and told us about the long, often over 50 mile, cycle run he had enjoyed on Sunday.The headmaster, Mr Bernard Johnson, described[Hartley]as "a boy of most unusual ability and power of concentration" and said that he did not remember "any pupil in over30years' teaching experience who was superior to him. His results in the General Certificate were phenomenal, easily a record for the school." Like many boys good at mathematics, Brian also excelled at chess. Usually top of the school chess table, he became captain of the school chess team, won local championships and played for the Lancashire team. He was one of the school's most enthusiastic musicians; this early love showed itself principally in trumpet playing and jazz.

His results in the General Certificate of Education mentioned in the above quote were Advanced Level in Mathematics, Physics and Chemistry all passed with distinction in 1956. Ian Stewart, one of his doctoral students, described Hartley's school years [5]:-

In fact Hartley was awarded two scholarships to attend university, a State Scholarship based on his outstanding results in the General Certificate of Education Advanced Level papers, and a Major Scholarship in Natural Sciences at King's College, Cambridge, won in examinations taken in December 1956. He could have either read mathematics or natural sciences at this stage but, after being invited to King's College to discuss his options with Albert Ingham, the Director of Studies in Mathematics at King's, he made the choice that mathematics was for him and in October 1957 he matriculated at Cambridge. At this stage Hartley entered a Mathematics Faculty which was full of world-leading researchers. There was Abram Besicovitch, who retired one year after Hartley began his studies and was replaced as Rouse Ball Professor of Mathematics by Harold Davenport. There was Philip Hall at King's College who, by that time, held the Sadleirian Chair but did not supervise undergraduates. In addition there was William Hodge who was Lowndean Professor of Astronomy and Geometry. Also lecturers in the Mathematics faculty at this time were Frank Adams, Michael Atiyah, J W S (Ian) Cassels, Peter Swinnerton-Dyer, John Arthur Todd and Chris Zeeman.Holiday jobs included working in a pie factory and a cotton mill, where he had to jump into tubs of raw cotton to pack them down. At15, he became one of the youngest people to be awarded a scholarship to King's College, Cambridge. Arriving as a17-year-old he planned to study chemistry but after attending research lectures given by the mathematician Philip Hall he switched.

Hartley excelled in the Cambridge Mathematical tripos, taking Part I in 1958 and Part II in 1960. He studied hard but also found time to keep up his sporting interests. He was Secretary of the Cambridge University Cycling Club, representing the Club in its annual competition with Oxford University in 1960, 1961 and 1962. In the summer of 1960, after taking the Part II examinations, Hartley went cycling in Scandinavia with a group of friends, cycling to the Artic Circle and back. At a dance at Christmas 1960, Hartley met Mary E Mawer. They were married in the summer of 1962 and, at this time, with Hartley living in Memorial Court, Clare College, Cambridge, the Master of Clare offered them the Master's Lodge for their reception. Brian and Mary had two children, Catherine Frances Hartley (born in January 1972) and Christine Elaine Hartley (born in April 1974). Let us return to pick up the story of Hartley's Cambridge career.

In June 1961 Hartley took the examinations for Part III of the Mathematical tripos. He was awarded distinction, as was his fellow student Trevor Hawkes who later became a colleague and is one of the authors of [3]. He then began research on group theory advised by Philip Hall. In December 1961, while only beginning to undertake research, Hartley was elected to a Research Fellowship at Clare College. As a fellow, he lived in Memorial Court, as we noted above. After he was married in the summer of 1962, Brian and Mary moved into a house on Blinco Grove on the outskirts of the city close to Homerton College. Hartley's 127-page Ph.D. thesis *The stability group of a series of subgroups* was submitted in 1964 but before that, in May 1963, he had submitted a paper jointly written with Philip Hall, with the same title as his thesis to the London Mathematical Society. The paper was revised in August 1965 and published the following year. Karl Gruenberg wrote in a review of the paper:-

In fact Hartley's first published paper wasThere is a wealth of interesting and illuminating detail.

*The order-types of central series*which appeared in 1965. Kurt Hirsch wrote in a review that:-

Hartley spent the year 1964-65 in Chicago which at that time was a world centre for algebra with many leading ring theorists on the staff such as Adrian Albert, Yitz Herstein, Irving Kaplansky, who was the departmental chairman, and Saunders Mac Lane. Although in a different area of group theory from Hartley, John Thompson was also at Chicago and had just gained world fame with his 1963 paper, written with Walter Feit, proving all nonabelian finite simple groups were of even order. During the year in Chicago, Hartley enjoyed the music scene, both classical and jazz, and continued his love of the outdoors learning to ski and going on camping holidays. He spent the following year at the Massachusetts Institute of Technology where he was appointed as C L E Moore Instructor.All the proofs are given by explicit constructions, some of them quite ingenious.

Returning to England, he was appointed to the University of Warwick in 1966. This was a new university which took in its first undergraduates in October 1965. On the staff were Chris Zeeman, the head of the Department of Mathematics, David Epstein, Roger Carter, Sandy Green, Luke Hodgkin, Brian Sanderson and Ralph Schwarzenberger. I [EFR] became a postgraduate student at Warwick in October 1965 studying for an M.Sc. advised by Roger Carter. Sandy Green was given the task of recruiting algebraists onto the staff for the academic year beginning in October 1966 and, after taking advice from Philip Hall, invited Stewart Stonehewer, Trevor Hawkes and Brian Hartley. All accepted and Stonehewer and Hartley took up their appointments for the academic year 1966-67. With these new appointments it was an even more exciting place for an algebraist and I continued to study for a Ph.D. advised by Stewart Stonehewer. I submitted my Ph.D. thesis *Classes of Generalised Nilpotent Groups* in 1968 and Hartley was appointed as the internal examiner. I remember my Ph.D. oral when Brian said he had proved a number of the same results that I had in my thesis but, since he had never got round to publishing them, I couldn't be expected to know that. He was a wonderful examiner, looking to see how I could develop the ideas of my thesis further, being kind and positive. After the oral, Brian invited me to dinner and I spent a very enjoyable evening with him and Ian Stewart, Brian's first Ph.D. student.

Except for spending 1969-70 at the Australian National University in Canberra, Australia and 1974-5 in Madison, USA, at the University of Wisconsin, Hartley remained at Warwick until he was appointed to a chair in Manchester in 1977. His only book *Rings, modules and linear algebra* (written with T O Hawkes), written in his early years at Warwick and published in 1970, became a widely used undergraduate text. You can read extracts from three reviews of this book at THIS LINK.

Although Hartley was appointed to the University of Manchester in 1977, he had been one of the two main organisers of the 1977-78 Warwick Symposium on Infinite Groups and Group Rings. He spent much time at Warwick during this year and again I [EFR] was able enjoy his company since I took sabbatical leave in 1977-78 and spent most of the year at the Symposium. It wasn't until the spring of 1978 that Hartley and his family moved to Didsbury, about 5 km south of the University of Manchester. This was an excellent location for Hartley who cycled daily to the University from his home along fairly minor roads. His contribution to the University of Manchester was a major one. He organised a weekly algebra seminar, brought many leading algebraists to Manchester as visitors, and supervised many Ph.D. students.

He continued with his love of travelling and spent time at the University of Utah in 1981-82, the University of Singapore in 1985, and the University of Minia, Egypt, in 1984 and 1985. In 1989 he visited the Mathematical Sciences Research Institute, Berkeley, California at Easter, Padua and Moscow in the summer, and East Berlin and Barcelona in the autumn.

His over 100 publications are almost all on group theory. If group rings are considered a part of group theory, then Hartley only wrote two papers which were not on group theory, one being on Lie algebras and the other on cryptography and large primes. His main topic was locally finite groups where he used his wide knowledge of finite groups in proving properties of infinite groups which were in a sense close to finite. A locally finite group is a group in which every finite set of elements is contained in a finite subgroup. The study of locally finite groups began with Issai Schur's result that a periodic linear group is locally finite. He collaborated with many mathematicians and loved working out ideas with them on a blackboard. In fact he wrote papers with over 50 different mathematicians.

Roger Bryant, a colleague at Manchester, described Hartley as [5]:-

Hartley's interests outside mathematics, and there were many, are described by Ian Stewart as follows [5]:-... a straightforward personality. Talked directly with everyone. No airs and graces: taxi-driver, research student, whatever. However, he stuck to what he believed in.

The authors of [3] write:-Brian was also a musician. As a teenager he learned the trumpet. He played in St Catherine's Military Band, Accrington, and Cambridge University first Orchestra; he once performed at a mathematics conference but as he had become a little rusty, part way into the piece he and his accompanist declared they had merely been tuning up. They started again, with greater success. Brian was an outdoor person. In his younger days he ran with Ron Hill, who later competed in several Olympic marathons. The Hartleys owned a series of tandems, but Mary[his wife]tended to fold her arms and sing as they went up hills, leaving Brian to do all the work. They both took up cross-country skiing, ice-skating, and horse-riding. The whole family enjoyed camping. Above all, Brian and Mary delighted in hill walking, which occupied most weekends.

Let me [EFR] add a comment of my own. Brian was a very practical person who was good with his hands. If the washing machine broke down, then he would mend it. When I expressed my surprise, Brian explained that one could do anything, "just get a book that tells you how to do it," he said. This positive attitude was not only good for mending washing machines, it must have contributed to his mathematical success too. The last time I met with Brian was at the 'Infinite groups 1994' conference held in Ravello, Italy, 23-27 May 1994. Ravello is a stunningly beautiful town on the Amalfi coast and Brian loved being there. He gave the talkHis intellectual energy, enthusiasm for algebra, direct manner and dry sense of humour endeared him to the many mathematical friends he made around the world. He was devoted to mathematics and gave generously of his time and energy in support of younger colleagues.

*Locally finite groups whose Sylow subgroups are Chernikov*at the conference and his main aim in this talk was to describe results obtained by S D Bell at the University of Manchester. Brian was the chair of the session in which I delivered my paper and he discussed it with me afterwards, as always making helpful suggestions. In Ravello, Brian told me about the NATO Advanced Study Institute on Finite and Locally Finite Groups he was helping to organise in Istanbul, Turkey, which was be held in August of that year. He was excited about his trip to Istanbul, and also about the conference with around 100 participants he was expecting from many countries. It was a great shock to me to hear of Brian's death just over a month after he returned from the Istanbul conference.

In fact Brian died walking in the hills that he loved. There was a mountain in northern England that Brian always wanted to climb, namely Helvellyn, and he decided to try it before he got too old. He did make the top but collapsed with a heart attack on the way down.

The picture of Brian Hartley above was taken in August 1993 at the Groups Galway / St Andrews Conference, held in Galway, Ireland.

**Article by:** *J J O'Connor* and *E F Robertson*