Graphs defined on groups LTCC Intensive Course, 89 June 2021 
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I am a halftime Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London. In addition, I am an associate researcher at CEMAT, University of Lisbon, Portugal.
I am a Fellow of the Royal Society of Edinburgh.
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Diagonal groups D(G,m) for G a finite simple group are one of the classes arising in the O'NanScott Theorem. However, they can be defined for any group G, not necessarily finite or simple. With Rosemary Bailey, Cheryl Praeger and Csaba Schneider, I have defined for each pair G,m where G is a group and m a dimension greater than 1, a geometric structure called a diagonal semilattice whose automorphism group is the corresponding diagonal group. We also gave simple axioms for these geometries. For m = 2, a structure satisfying the axioms is precisely a Latin square, but for m > 2 the group G arises naturally from the combinatorial axioms.
These structures consist of m+1 partitions of a set, any m of which are the least elements in a Cartesian lattice. With Rosemary, Cheryl, and Michael Kinyon, I have looked at larger sets of partitions satisfying these conditions, and found connections with orthogonal arrays, arcs in projective space, and MDS codes.
Finally, the paper with Sean Eberhard contains a mistake which we have been unable to fix.
There are various classes of graphs which are recognised by a small list of forbidden induced subgraphs, and which have nice algorithmic properties and various applications; these include perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs. With Pallabi Manna and Ranjit Mehatari, I have investigated when the power graph of a group belongs to one of these classes. Power graphs are always perfect; we have found all groups for which they are either split or threshold, and all nilpotent groups for which they are either cographs or chordal. In addition, there is a long survey article on power graphs. One curious fact proved here is that, if f(n) is the clique number of the power graph of a cyclic group of order n, then φ(n) ≤ f(n) ≤ cφ(n), where φ is Euler's totient function and c = 2.6481017597….
In two papers with Collin Bleak and Shayo Olukoya, I have looked further at the connection between strongly synchronizing automata and transducers, the Higman–Thompson finitely presented infinite simple groups, and the automorphism group of the one or twosided shift.
Old research snapshots are kept here.
I am Honorary EditorInChief of the Australasian Journal of Combinatorics, an international openaccess journal published by the Combinatorial Mathematics Society of Australasia. 
School of Mathematics and Statistics
University of St Andrews North Haugh St Andrews, Fife KY16 9SS SCOTLAND 
Tel.: +44 (0)1334 463769 Fax: +44 (0)1334 46 3748 Email: pjc20(at)starthurs(dot)ac(dot)uk [oops – wrong saint!] 
Page revised 29 August 2021 
A group G is said to satisfy the finiteness condition F_{n} if it acts geometrically (that is, properly and cocompactly) on an (n−1)connected CWcomplex. It satisfies F_{∞} if it satisfies F_{n} for all n. Here, F_{1} is equivalent to being finitely generated, and F_{2} to being finitely presented. Finitely generated free groups satisfy F_{∞}.
The Houghton group H_{n} satisfies F_{n} but not F_{n+1}. Also, this group contains the finitary symmetric group, so it is highly transitive.
Problem: Given a natural number n, does there exist a group G such that
Old problems are kept here.