# The beauty of symmetry

Symmetry is everywhere, at every scale. It appears in the construction of buildings, in nature in spiral shells and plants, and in space as spiral galaxies.

Many problems faced by engineers, scientists and mathematicians use group theory and symmetry to help solve them. Group theory uses symmetries of a complex system to make difficult problems more manageable. It is used to measure symmetry and is used in technologies from solving Rubik’s cube to internet communications and secure banking.

Artists and mathematicians notice it. It is my passion, mathematics and symmetry.

Prof. Cheryl Praeger AM FAA, UWA

### A mathematical mind

Emeritus Professor Cheryl Praeger AM FAA is a multiple award-winning professor with a passion for mathematics. She has published over 410 scientific papers and her work is highly respected internationally.

Her work is complex with key research in the field of group theory, particularly permutation groups and combinatorics, a branch of pure mathematics which studies symmetry. Her work has opened up large parts of mathematics and provides tools and theories that can be applied to a wide range of problems.

For over 40 years, Emeritus Professor Praeger has been dedicated to the pursuit of mathematical knowledge. She is part of the internationally recognised team of pure mathematicians at The Mathematics of Symmetry and Computation Research Cluster at The University of Western Australia (UWA). Her work has been influential in many academic and real world applications.

Maths holds answers to almost anything we want to do in life.

Prof. Cheryl Praeger AM FAA, UWA

### Applying a mega theorem

In 1979, the mega theorem Finite Simple Group Classification identified the mathematical building blocks of symmetry. It paved the way for mathematicians, including Professor Praeger, to understand symmetric structures in nature, in science and in mathematics, in novel ways. It spanned across algebra and combinatorics and changed the problems that could be solved and the methods used to solve them.

Still early in her career, Professor Praeger and her team were among the first to apply the classification to group theory and permutation groups.

Everything we did required new fundamental theory to be developed.

Prof. Cheryl Praeger AM FAA, UWA

- In 1983, solving the long standing Sims conjecture, a powerful tool in group theory. Her explanation has been widely cited by researchers.
- In 1988, proving an influential theorem in permutation group theory: the O’Nan-Scott Theorem. She recognised that the real power was in the theorem’s ability to split the finite primitive groups into various types.
- In 1990, producing a ‘Factorisations memoir’. The detailed study of the geometry of simple groups is widely used by mathematicians and scientists working with finite simple group classification.
- In 1993, pioneering the theory of quasi-primitive permutation groups. The theory changed the face of group theory and has driven work in algebraic applications of graph theory. It has become a widely accepted tool used by mathematicians around the world.

#### Taming the Monster

The largest of the sporadic simple groups (the bottom right square in the Periodic Table of Finite Simple Groups) is called the Monster. It contains 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements (more than 8 followed by 53 zeros), and has fascinated scientists for decades, since its construction in the 1980s. It has even been speculated by Princeton physicist and Fields Medallist Edward Witten that the Monster might be associated with quantum gravity and might even be the symmetry group of a black hole.

In the team’s Memoirs of the American Mathematical Society book: Automorphism orbits and element orders in finite groups: almost-solubility and the Monster (yet to be published, Nov 2020), the team study how “homogeneous” (in some sense, how “symmetric”) the finite simple groups can be. They introduce a parameter which measures the homogeneity, and prove that the Monster is uniquely the most homogeneous simple group. The Austrian Science Fund supports the book’s first author, Dr Alexander Bors. The second and third authors, Professor Michael Giudici and Emeritus Professor Cheryl E. Praeger are supported by the Australian Research Council as part of a Discovery Project.

### Super powerful algorithms

Mathematics is behind all the algorithms that make the computers work.

Professor Praeger’s innovative research into symmetry has produced efficient algorithms which are used in far-reaching applications including search engines to retrieve information from large networks such as the world wide web.

It’s like space travel. You find a worm hole that takes you to the other side, when you haven’t really spent the time to get there.

Prof. Cheryl Praeger AM FAA, UWA

She and her team developed algorithms which compute with matrix groups. Matrix groups occur in almost every area of mathematics to describe the symmetry of mathematical and technological systems, as well as in many areas of science. The algorithms enabled users to make giant leaps in decision making problems and have transformed the way in which algebra is researched and taught.

The Neumann-Praeger SL recognition algorithm was developed in 1992 and became influential in launching the international Matrix Group Recognition Project. Still in progress the project produces algorithms for efficient matrix group problems and looks for ways in which to implement them.

Many of her algorithms have had real world impact and have also been built into the GAP and MAGMA computer systems. These systems are used by mathematicians, scientists and engineers world-wide for research and teaching.

We realised that the symmetry of the designs was important. I love precision and like to know that my predications of performance are really matched in practice

Prof. Cheryl Praeger AM FAA, UWA

### Making crops count

In the early 1980’s, Professor Cheryl Praeger worked on the use of combinatorial design layouts in agricultural experiments for crop production.

The team produced a uniform rigorous analysis (of variance) for a much larger class of experimental layouts than had previously been considered, using group theory. By doing this, they could minimise the effect of other influences such as climate, field position, position of the sun, and rainfall to produce a manageable and reliable inferences about the data.

Agricultural statisticians could then use the data to compare the yields of different crop varieties to understand which crops would perform better in different situations. Her analysis shaped the courses that were taught to statisticians that would go on to become agricultural research developers. These agricultural trials have been important in transforming the value of the agricultural industry and economy in producing higher yields and health across the world.

### The mathematics of weaving

In 1982, whilst on maternity leave, Prof Praeger contributed to one of her most popular lecture topics. She was approached by colleagues in computer science who were looking at the specific design sequence of ‘overs and unders’ that make up twill; a family of fabrics that have important industrial properties.

The team applied group theory, producing a series of papers and complex matrix equations and algorithms to respond to the problems in the weaving process such as whether or not the woven fabric would lie flat or tend to curl at the edges, and controlling how far threads could float before being woven back into the fabric. The mathematical model and calculations gave a precise count of the total number of twills, with pre-assigned balance and float length properties, and underpinned an accurate enumeration of the possible twills, that would go on to aid weavers in the sorts of patterns they could create, as well as aid them in deciding how to tie up the loom and weave.

It’s like a superpower. Using powerful mathematical tools allows you to produce simple results, yet what is underpinning them is so complicated.

Prof. Cheryl Praeger AM FAA, UWA

### An inspirational teacher and mentor

Professor Praeger has helped develop mathematical projects promoting the importance of mathematics globally including in emerging countries.

She has also supervised a number of international and local postgraduate students. Among her students was Professor Akshay Venkatesh, **winner of the 2018 Fields Medal**. Her passion for mentoring and her work in developing mathematics in Australia, earned her recognition from the Australian Government in 2015 for “developing the gold standard in mathematics research supervision”.

I’m so lucky that my career has meant that I have a greater influence on mathematics. I also have a great passion to support developing countries.

Prof. Cheryl Praeger AM FAA, UWA

Professor Praeger started at UWA in 1976. She was appointed Professor in 1983 at age 35, whilst a mother of two pre-school children. She was only the second woman professor of Mathematics at an Australian University. In the early years of her appointment, Professor Praeger introduced computers into the curriculum to modernise the course, and to teach the new generation of maths graduates with these new skills.

She was one of the first mathematicians in Australia to develop a project that attracted research funding. Funding encouraged international research visitors to UWA, fostering collaborative research and changing the way mathematics research could be conducted.

During her prestigious career, Professor Praeger has been awarded with distinguished *awards* to awards page and has held many positions on programs and committees in Australia and internationally. For her extensive service, and in particular for her fundamental work in group theory and combinatorics, Professor Praeger was awarded the 2019 Prime Minister’s Prize for Science.

### Counting on the next generation

Considered one of the most successful academics in the southern hemisphere, Professor Praeger has fostered a new generation of mathematicians. Heavily involved in the direction of mathematics within *Australia* to awards page, she has particularly promoted and supported girls and women in mathematics.

In 2004, Prof Praeger wrote The Essential Elements of Mathematics. This paper was originally written in response to an invitation from the Victorian Curriculum and Assessment Authority with respect to its work on developing a Framework of Essential Learning relating to the compulsory years of schooling. The views expressed in the paper are those of the author, and do not necessarily represent the views of the Authority.

Young people are the future of mathematics. One of the most rewarding parts of my job is seeing them find new discoveries.

Prof. Cheryl Praeger AM FAA, UWA

From its beginnings at UWA in 2003, the WA Junior Mathematics Olympiad (WAJO) has aimed to search for the most gifted students in mathematics. WAJO has grown to include over 400 children, with Professor Praeger overseeing the Olympiad annually.