Since a group is a measure of symmetry, group theory has found many applications in the sciences.
For example, in the 1960s, when physicists were discovering subatomic particles, Murray Gell-Mann, the American physicist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles, used group theory to develop a theory for the mesons and baryons. These were encoded through a group representation and each dimension corresponded to a particle. There were eight mesons and these formed an eight-dimensional representation. However, the theory said that the representation for the baryons should be 10-dimensional yet they only knew of nine particles. This led him to predict a 10th and sure enough it was found several years later.
Group theory is also used in crystallography and knowledge of the possible symmetry groups enables the two-dimensional images from x-ray diffraction to be used to determine the three-dimensional structure of crystals, and led to the British scientist Francis Crick discovering the double helix structure of DNA in 1953.
Group theory and symmetry have also been used to study viruses.
Members of the centre undertake research into the fundamental properties of permutation groups and use the developed theory to study classes of mathematical objects with a high degree of symmetry, such as graphs and geometries.
Some highlights in permutation group theory include:
Luke Morgan and Michael Giudici developed a structure theory for semiprimitive groups, that is for transitive groups for which every normal subgroup is either transitive or semiregular. Semiprimitive groups are a generalisation of primitive and quasi primitive groups and play an important role in questions about local actions on graphs. Cheryl Praeger had previously developed a characterisation theorem for the class of quasiprimitive groups, that is, those groups for which every nontrivial normal subgroup is transitive. This class of groups includes all primitive groups and the characterisation is a generalisation of the O'Nan-Scott Theorem for primitive groups. The characterisation theorem has driven a large body of work in algebraic graph theory.
A sequence of three papers involving John Bamberg, Michael Giudici and Cheryl Praeger along with co-authors Martin Liebeck, Jan Saxl and Pham Huu Tiep, classified all 3/2-transitive permutation groups. Wielandt had shown in the 60's that any such group is either Frobenius or primitive and few nontrivial example were known. Paper 1, Paper 2, Paper 3.
A collaboration between Joanna Fawcett, Michael Giudici, Cai Heng Li, Cheryl Praeger, Gordon Royle and Gabriel Verret initiated at one of our annual research retreats classified all primitive permutation groups with a suborbit of length 5. Those with suborbits of length 3 or 4 had previously been classified by Wong, Sims, Quirin and Wang. The classification was then used to classify all vertex-primitive graphs of valency 5 and prove that the smallest valency of a half-arc-transitive primitive graph is 12.
Together with Martin Liebeck and Jan Saxl, Cheryl Praeger has achieved a classification of all regular subgroups of primitive almost simple groups. This work relies heavily on their earlier classification of the maximal factorisations of almost simple groups. Previously, Cai Heng Li had given a complete classification of the finite primitive permutation groups containing a regular abelian subgroup.
Michael Giudici has shown that apart from the wreath products M wr K acting on 12 points, every primitive permutation group contains a fixed point free element of prime order. It had previously been shown by Fein, Kantor and Schacher that any transitive permutation group has a fixed point free element of prime power order, while Giudici's result and subsequent work makes substantial progress towards the Polycirculant Conjecture that every vertex-transitive graph has a semiregular automorphism.