Graphs defined on groups|
LTCC Intensive Course, 8-9 June 2021
|Files available here|
Welcome to my St Andrews homepage. This page is under construction (and probably always will be!)
I am a half-time 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.
On this site
Diagonal groups D(G,m) for G a finite simple group are one of the classes arising in the O'Nan--Scott 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 two-sided shift.
Old research snapshots are kept here.
|I am Honorary Editor-In-Chief of the Australasian Journal of Combinatorics, an international open-access journal published by the Combinatorial Mathematics Society of Australasia.|
School of Mathematics and Statistics
University of St Andrews
St Andrews, Fife KY16 9SS
Tel.: +44 (0)1334 463769
Fax: +44 (0)1334 46 3748
[oops – wrong saint!]
Page revised 29 August 2021
A group G is said to satisfy the finiteness condition Fn if it acts geometrically (that is, properly and cocompactly) on an (n−1)-connected CW-complex. It satisfies F∞ if it satisfies Fn for all n. Here, F1 is equivalent to being finitely generated, and F2 to being finitely presented. Finitely generated free groups satisfy F∞.
The Houghton group Hn satisfies Fn but not Fn+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.