**Cheryl Praeger**'s parents, Eric Noel Praeger and Queenie Hannah Elizabeth Praeger, were both from a poor background and, having to leave school to earn their living, had no chance of participating in higher education. Both Cheryl's parents, however, encouraged their children to educational achievements. Cheryl's mother always had an ambition that her children would be able to benefit from a university education. When Cheryl was born her father was working for the Commercial Bank of Australia and this led to the family moving from one town to another as Cheryl was growing up. In an interview with Bernhard Neumann in 1999, Cheryl said [4]:-

We lived on the Darling Downs -- in Toowoomba, where I was born, and then Warwick. I was about to start school in Warwick when he was transferred to Margate, on Moreton Bay just north of Brisbane, so I began school there at Humpybong State School -- a wonderful name. After four years we moved again, to Nambour, about60miles north of Brisbane. By that stage Dad was an accountant in the bank. He underwent manager training and expected to be moved as manager of a branch somewhere, but I think the new manager in Nambour wanted him to stay longer to help him settle in.

After one year at high school in Nambour, Cheryl's family moved again, this time to Brisbane, because her father had changed career and wanted to open an office there as a chiropractor. In Brisbane she attended the Brisbane Girls Grammar School as a day pupil for three years. By this time she knew that she wanted to study mathematics, but it might have been music as she explained [4]:-

My mother's youngest brother, Uncle Darcy, played the organ at a central Brisbane Baptist Church that we went to whenever we visited my Grandma. I could see him up in the organ loft at the front of the church, and I think I always wanted to play the piano. So my mother put away the money that the government used to give mothers in those days -- it was to buy us milk, I think -- and when I was eight years old she was able to buy a piano and I could begin to learn to play it. ... I thought about taking up music seriously, but at age11-- because of a certain amount of disobedience -- I had an accident in a pool and dislocated my finger and fractured it in a couple of places. It took two years of very serious physiotherapy for me to get it strong enough and straight enough to be able to play the piano again. So it was my ambition to play the piano but I certainly couldn't have taken it up professionally.

After leaving school Praeger enrolled in the University of Queensland and completed her four year B.Sc. degree in 1969. Her honours courses were entirely mathematics, but in her first two years of study she also took courses in physics. Between her third and fourth years Praeger spent eight weeks at the Australian National University. She had been awarded a scholarship to enable her to work there, obtaining her first experience of research in mathematics. Bernhard Neumann suggested a problem to her which she solved and so published her first paper *Note on a functional equation* while still an undergraduate. It was her first taste of research and [4]:-

It was a great thrill.

Praeger had studied the functional equation *x*(*n*+1) - *x*(*n*) = *x*^{2}(*n*), where *x*^{2}(*n*) = *x*(*x*(*n*)) and *x* is an integer-valued function of the integer variable *n*, and found a three-parameter family of solutions.

After completing her undergraduate course Praeger was offered a scholarship to undertake research at the Australian National University but she was also offered a Commonwealth Scholarship to study at the University of Oxford in England. She chose the latter. At Oxford, Praeger was in St Anne's College and assigned Peter Neumann (son of Bernhard Neumann and Hanna Neumann) as a supervisor for her research. She was awarded an M.Sc. in 1972 and then her doctorate in 1974 for her thesis *Finite Permutation Groups.* In fact she finished her doctorate in 1973 and returned to Australia in that year before being awarded her doctorate in the following year 'in absentia'. Her first paper on group theory *Sylow subgroups of transitive permutation groups* appeared in print in 1973

Back in Australia, Praeger was appointed to a three year postdoctoral fellowship at the Australian National University (ANU) but spent one semester at the University of Virginia in the United States. She attended the Second International Conference on the Theory of Groups at ANU in 1973 giving a talk on *Sylow subgroups of finite permutation groups* which was published in the Proceedings of the Conference. A series of three papers on a similar topic *On the Sylow subgroups of a doubly transitive permutation group* appeared in 1974 and 1975. At ANU Praeger lived at University House and there she met a statistics research student John Henstridge. They were married in August 1975. In early 1976 they moved to the University of Western Australia in Perth where Praeger had been offered a temporary two-year position and her husband was offered a one year tutorship which would allow him to complete his doctorate. They lived in St George's College, the oldest college at the University. Before the end of her contract, Praeger was offered a permanent post by the University of Western Australia which she chose in preference to a three-year lectureship she had been offered at the University of Melbourne [4]:-

In1978we moved from the college to a small unit, and our first child, James, was born in '79. Then our second son, Timothy, came in1982and we moved again -- the unit was just a bit small for us. We couldn't close the door because of the cot. Our house now is lovely ...

In fact Praeger wrote one joint paper with her husband, *Note on primitive permutation groups and a Diophantine equation,* which was published by the journal *Discrete Mathematics* in 1980. In 1982 Praeger was promoted to Senior Lecturer, then to Professor of Mathematics less than a year later.

Praeger has one of the most stunning publication records of any mathematician. Up to the time of writing this biography in September 2006 over 250 papers had appeared in print. This vast publication record makes it impossible to even give an overview of all the topics she has studied but we should remark on just a few to at least give a flavour of her remarkable contributions. We have already noted that her doctoral work was on permutation groups and her early papers were on this topic. Later this interest broadened to a more general study of group actions. From there she became interested in the structure of the objects on which the groups acted, so she moved into the theory of combinatorial designs. But this interest in designs also took another form [4]:-

I also became involved in designs used for experimental layouts for agricultural experiments that statisticians would analyse - to help statisticians to understand what symmetry groups were involved in the particular experimental designs which they were interested in.

Examples of papers in which Praeger looked at groups acting on structures are *Symmetric graphs and a characterization of the odd graphs* (1980) which investigates graphs with large groups of automorphisms. In *Enumeration of rooted trees with a height distribution* (1985) written jointly with P Schultz and N C Wormald, the authors used generating functions to find a new solution to the problem of determining the number of rooted trees whose vertices have a given height distribution.

Another topic to which Praeger had made major contributions is Computational Group Theory. She moved into this topic through first wanting to introduce computers into undergraduate teaching, then realising that having a research student in this area would help her develop an appreciation of computing. Having organised a workshop given by experts in computational group theory so that members of her research team at the University of Western Australia could gain expertise, she was told of areas which needed improvement. After making a strong contribution to solving these problems, she was invited to the Computational Group Theory meeting in Oberwolfach, Germany, in 1988. Joachim Neubüser, at that time the leader of the team developing the computing system GAP (Groups, Algorithms, and Programming), suggested at that meeting that an important area to develop would be finding algorithms to recognise matrix groups analogous to those which had already been invented for permutation groups. Praeger was immediately interested and she began an important line of research which has made major contributions to Computational Group Theory.

One of Praeger's most popular lecture topics at a lower level is on mathematics and weaving. She has published three papers on the topic: *Mathematics and weaving* (1986); *Mathematics and weaving. *I*. Fabrics and how they hang together* (1987); and *Mathematics and weaving. *II*. Setting up the loom and factorizing matrices* (1988).

Praeger has received many honours for her mathematical contributions. For example she was President of the Australian Mathematical Society during 1992-94, elected a fellow of the Australian Academy of Science in 1996, and made a Member of the Order of Australia in 1999.

In 2003 she was awarded the Centenary Medal of the Australian Government for services to Mathematics. In the following year she was selected for being in the top 1% of most highly cited research mathematicians worldwide, having published more than 250 research articles and 3 books. In 2005 the Université Libre de Bruxelles, Belgium, awarded her an honorary mathematics doctorate. She was given an honorary degree by the University of St Andrews in November 2015.

We have mentioned above some of Praeger's popular mathematics lectures. Let us give a recent example of a talk she gave on Tuesday 13 May 2008 to the 'Friends of the University Library' at the University of Western Australia. The talk was entitled *What makes a "good" mathematician?* and her abstract reads:-

Just about every part of modern society is underpinned by mathematical technology, and the only way to benefit from and manage the information/data explosion is by building smarter mathematics. But how smart are mathematicians? How can we measure what makes a mathematician "good"?

Finally let us note that among her hobbies are hiking, cycling, spinning wool, and sailing.

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

**September 2009**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Praeger.html]