## Research

### Meetings

My main area of research is in group theory. I am interested in simple groups, both finite and algebraic, with a particular focus on subgroup structure, conjugacy classes and representation theory. I am also interested in permutation groups and related combinatorics, and in the application of probabilistic and computational methods.

### Bases

Let $G \leqslant {\rm Sym}(\Omega)$ be a permutation group on a set $\Omega$. A subset $B \subseteq \Omega$ is a base for $G$ if the pointwise stabilizer of $B$ in $G$ is trivial. The base size of $G$, denoted by $b(G)$, is the minimal cardinality of a base. Bases have been studied since the early days of group theory in the 19th century, and in more recent years they have played an important role in the computational study of finite permutation groups. Determining the base size of $G$ is a fundamental problem in permutation group theory.

Let $G \leqslant {\rm Sym}(\Omega)$ be an almost simple primitive permutation group (recall that $G$ is almost simple if $G_0 \leqslant G \leqslant {\rm Aut}(G_0)$ for some non-abelian simple group $G_0$, and $G$ is primitive if $G$ is transitive and a point stabilizer is a maximal subgroup of $G$). Bases for such groups have been widely studied in recent years, and further interest stems from the following conjecture of Cameron and Kantor from the early 1990s:

Conjecture. There exists an absolute constant $c$ such that $b(G) \leqslant c$ for every non-standard permutation group $G$.
Here an almost simple primitive group $G \leqslant {\rm Sym}(\Omega)$ with socle $G_0$ is standard if $G_0=A_n$ is an alternating group and $\Omega$ is an orbit of subsets or partitions of $\{1, \ldots, n\}$, or if $G_0=Cl(V)$ is a classical group and $\Omega$ is an orbit of subspaces (or pairs of subspaces) of the natural module $V$. Otherwise, $G$ is non-standard. In general, the base size of a standard group can be arbitrarily large (for example, it is easy to see that $b(G)=\dim V +1$ for the action of $G={\rm PGL}(V)$ on $1$-dimensional subspaces of $V$), so the non-standard hypothesis is necessary.

The conjecture was proved by Liebeck and Shalev, for some undetermined constant $c$, using probabilistic methods based on fixed point ratio estimates. Later, in a series of papers with various co-authors (Guralnick, Liebeck, O'Brien, Saxl, Shalev and Wilson), we proved that $c=7$ is optimal:

Theorem. Let $G$ be a non-standard permutation group. Then $b(G) \leqslant 7$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its 5-transitive action of degree 24.
This confirms a conjecture of Cameron - see the following paper, and the references therein:
T.C. Burness, M.W. Liebeck and A. Shalev
Base sizes for simple groups and a conjecture of Cameron
Proceedings of the London Mathematical Society 97 (2009), 116-162
In the same paper, we also show that if $G \leqslant {\rm Sym}(\Omega)$ is a non-standard group then the probability that 6 randomly chosen points in $\Omega$ form a base for $G$ tends to 1 as the order of $G$ tends to infinity.

A major ongoing project is to determine the exact base size of all non-standard almost simple primitive groups. This problem has been solved for alternating and sporadic groups. For groups of Lie type, substantial progress has been made through joint work with R. Guralnick (University of Southern California) and J. Saxl (University of Cambridge). For example, in the following paper we determine the exact value of $b(G)$ when $G$ is an almost simple primitive classical group with point stabilizer belonging to Aschbacher's collection $\mathcal{S}$ of almost simple irreducibly embedded subgroups.

T.C. Burness, R.M. Guralnick and J. Saxl
Base sizes for $\mathcal{S}$-actions of finite classical groups
Israel Journal of Mathematics 199 (2014), 711-756
This work is part of a wider study of bases for finite primitive permutation groups, which is an active area of current research. A long term goal is to determine all the primitive groups $G$ with the extremal property $b(G)=2$. This is an interesting problem, with connections to other areas of mathematics. For example, suppose $G=V \rtimes G_0 \leqslant {\rm AGL}(V)$ is a primitive affine group, so $G_0 \leqslant {\rm GL}(V)$ is irreducible. Then $b(G)=2$ if and and only if $G_0$ has a regular orbit on $V$; determining if a group has a regular orbit on an irreducible module is a classical problem in representation theory.

One can also study bases for infinite permutation groups. In recent work with Guralnick and Saxl, we have developed a new theory of bases for algebraic groups (over any algebraically closed field). Two new base measures arise naturally in this context (which we call the connected and generic base sizes) and in the following paper we calculate all three base measures for every primitive action of a simple algebraic group.

T.C. Burness, R.M. Guralnick and J. Saxl
On base sizes for algebraic groups
Journal of the European Mathematical Society (JEMS), to appear
These results at the level of algebraic groups have important implications for the finite primitive groups of Lie type that arise by taking the fixed points of a suitable Frobenius morphism. Indeed, our results for algebraic groups in the above paper are an important tool in our ongoing investigation of bases for finite almost simple groups.

Finally, let me mention one of the main open problems in this area. It is very easy to see that if $G$ is a permutation group of degree $n$ then $$\frac{\log|G|}{\log n} \leqslant b(G) \leqslant \log_2\!|G|.$$ However, a well known conjecture of Pyber from the early 1990s asserts that all primitive permutation groups have rather small bases:

Pyber's conjecture. There exists an absolute constant $c$ such that $$b(G) \leqslant c\frac{\log|G|}{\log n}$$ for every primitive permutation group $G$ of degree $n$.
It is easy to see that the primitivity hypothesis is necessary. This conjecture has received a fair amount of attention; the main approach has been to use the O'Nan-Scott theorem to reduce the conjecture to some specific families of primitive groups. Indeed, the conjecture has been verified for all almost simple and diagonal-type groups (through work of Benbenishty, Fawcett, Liebeck and Shalev), and certain affine-type groups have also been handled. In joint work with Á. Seress (Ohio State University), we have settled the conjecture for all non-affine groups.
Theorem. Pyber's base size conjecture holds for all non-affine groups.
The proof is highly combinatorial, and the details can be found in the following paper:
T.C. Burness and Á. Seress
On Pyber's base size conjecture
Transactions of the American Mathematical Society, 367 (2015), 5633-5651
Therefore, in order to complete the proof of Pyber's conjecture it remains to deal with primitive affine groups of the form $G=V \rtimes G_0 \leqslant {\rm AGL}(V)$, where $G_0 \leqslant {\rm GL}(V)$ is irreducible and preserves a non-trivial direct sum decomposition of $V$.

### Derangements

Let $G \leqslant {\rm Sym}(\Omega)$ be a permutation group on a set $\Omega$. A permutation $x \in G$ is a derangement if it has no fixed points on $\Omega$. The study of derangements has its origins in the early days of probability theory. For example, Pierre de Montmort investigated the probability of winning various games of chance that were popular in the salons and gambling dens of early 18th century Paris. In a highly influential book on this topic, published in 1708, he proved that the proportion of derangements in the symmetric group $S_n$ in its natural action on $n$ points is given by the formula $$\frac{1}{2!}-\frac{1}{3!}+\cdots +\frac{(-1)^{n}}{n!}$$ In particular, this proportion tends to $1/e$ as $n$ tends to infinity.

Assume $G$ is finite and transitive, with $|\Omega|=n \geqslant 2$. By the Orbit-Counting Lemma, $G$ contains a derangement. In the last 30 years, various extensions of this classical theorem of Jordan have been investigated. For example, Cameron and Cohen (1992) proved that the proportion of derangements in $G$ is at least $1/n$, with equality if and only if $G$ is sharply 2-transitive. Much more recently, a theorem of Fulman and Guralnick establishes the existence of an absolute constant $\epsilon>0$ such that the proportion of derangements in any transitive simple group is at least $\epsilon$.

It is also natural to consider the existence of derangements of a specified order. A theorem of Fein, Kantor and Schacher (1981) states that $G$ contains a derangement of prime power order (the proof relies on the Classification of Finite Simple Groups), and this result has important number-theoretic applications. However, not all transitive groups contain a derangement of prime order; the groups that do not contain such an element are called elusive groups, and they have been the subject of several papers in recent years. In particular, the primitive elusive groups have been determined by Giudici (2003).

In joint work with M. Giudici (University of Western Australia) and R. Wilson (Queen Mary) we have initiated a quantitative study of derangements of prime order in transitive permutation groups. Note that $G$ contains a derangement of prime order $r$ only if $r$ divides $|\Omega|$. Therefore, given a non-elusive group $G$ and a prime divisor $r$ of $|\Omega|$ it is natural to ask if $G$ contains a derangement of order $r$. Given such a prime $r$, we say that $G$ is $r$-elusive if $G$ does not contain a derangement of order $r$.

For primitive groups, we use the O'Nan-Scott theorem to essentially reduce the problem to almost simple groups, and we observe that Giudici's theorem implies that the only elusive almost simple primitive group is the 3-transitive action of the Mathieu group ${\rm M}_{11}$ of degree $12$. We then go on to determine all the $r$-elusive groups with socle an alternating or sporadic group. The details are given in the following paper:

T.C. Burness, M. Giudici and R.A. Wilson
Prime order derangements in primitive permutation groups
Journal of Algebra 341 (2011), 158-178
For groups of Lie type, Giudici and I have written a book on derangements of prime order in classical groups, which will be published in October 2015. Our analysis is based on Aschbacher's subgroup structure theorem, which provides a description of the maximal subgroups of classical groups. We also require detailed information on the conjugacy classes of elements of prime order in classical groups. The first half of the book aims to provide an accessible introduction to the finite classical groups, with a particular focus on their underlying geometry, conjugacy classes and subgroups. In the second half we give a detailed analysis of derangements of prime order in primitive classical groups.
T.C. Burness and M. Giudici
Classical groups, derangements and primes
Australian Mathematical Society Lecture Series, vol. 25
Cambridge University Press, 2016
ISBN: 978-1-1076-2944-8

In a different direction, H.P. Tong-Viet (University of Pretoria) and I have been investigating finite primitive permutation groups with extremal derangement properties. Let $G \leqslant {\rm Sym}(\Omega)$ be a finite primitive group and observe that the set of derangements in $G$ is a normal subset, so we can define $\kappa(G)$ to be the number of conjugacy classes of derangements in $G$. By Jordan's theorem, $\kappa(G) \geqslant 1$. In the paper

T.C. Burness and H.P. Tong-Viet
Derangements in primitive permutation groups, with an application to character theory
Quarterly Journal of Mathematics 66 (2015), 63-96
we prove the following theorem:
Theorem. Let $G$ be a finite primitive permutation group of degree $n$. Then $\kappa(G)=1$ if and only if $G$ is sharply $2$-transitive, or $(G,n) = (A_5,6)$ or $({\rm Aut}({\rm PSL}_{2}(8)),28)$.
In the same paper we also obtain detailed information on the primitive groups $G$ with $\kappa(G)=2$ (including a complete classification of the almost simple groups with this property) and we describe an interesting application to the character theory of finite groups.

Tong-Viet and I have also studied the finite primitive groups with the extremal property that every derangement in $G$ has $r$-power order, for some fixed prime $r$ (recall that the aforementioned theorem of Fein et al. guarantees the existence of a derangement of prime power order). Using the O'Nan-Scott theorem we first show that such a group is either almost simple or affine, and we completely determine all the almost simple groups. It turns out that the affine groups with this property have been extensively studied in earlier work of Guralnick and Wiegand (1992) and by Fleischmann, Lempken and Tiep (1997). The details can be found here:

T.C. Burness and H.P. Tong-Viet
Primitive permutation groups and derangements of prime power order
Manuscripta Mathematica 150 (2016), 255-291

### Generation and random generation

Let $G$ be a non-abelian finite simple group. It is well known that $G$ can be generated by two elements (the proof requires the Classification of Finite Simple Groups) and various extensions have been investigated in recent years. For example, a theorem of Liebeck and Shalev from 1995, which builds on earlier work of Dixon (1969) and Kantor and Lubotzky (1990), states that the probability that two randomly chosen elements generate $G$ tends to 1 as the order of $G$ tends to infinity. In a different direction, restrictions on the orders of a generating pair have also been widely studied. Note that the simple groups that can be generated by an element of order 2 and one of order 3 are particularly interesting; for instance, they correspond to quotients of the modular group ${\rm PSL}_2(\mathbb{Z}) = \mathbb{Z}_2\star \mathbb{Z}_3$.

Our understanding of the subgroup structure of finite simple groups has advanced greatly in the last 30 years or so. Indeed, many results on the generation of simple groups rely on detailed information on their maximal subgroups; the connection comes from the basic observation that if $x,y \in G$ do not generate $G$ then $\langle x,y \rangle$ is contained in a maximal subgroup of $G$.

In joint work with M. Liebeck (Imperial College) and A. Shalev (Hebrew University of Jerusalem) we have studied several generation properties of the maximal subgroups themselves, with the aim of extending some of the aforementioned results for simple groups to their maximal subgroups (with some suitable, and necessary, modifications). For instance, one of our main results is the following:

Theorem. Every maximal subgroup of a finite simple group can be generated by 4 elements.
This result is optimal in the sense that there are infinitely many examples that do indeed require 4 generators. New results on the random generation of maximal subgroups are also obtained, and we discuss applications to second maximal subgroups and primitive permutation groups. The details can be found in the following paper:
T.C. Burness, M.W. Liebeck and A. Shalev
Generation and random generation: from simple groups to maximal subgroups
Advances in Mathematics 248 (2013), 59-95

Many interesting properties of a 2-generated finite group $G$ are encoded in its generating graph, denoted by $\Gamma(G)$. Here the vertices are the non-identity elements of $G$, and two vertices are joined by an edge if and only if they generate $G$. For example, $G$ is $\frac{3}{2}$-generated if and only if $\Gamma(G)$ has no isolated vertices, and the "spread" of $G$ is at least 2 if and only if $\Gamma(G)$ is connected with diameter 2. The graph-theoretic viewpoint leads to many new and natural questions. For instance, one can investigate the connectedness of $\Gamma(G)$ (and subsequently its diameter), its (co-)clique and chromatic numbers, the existence of a Hamiltonian cycle in $\Gamma(G)$, and so on.

This is an active area of current research, especially in the context of simple groups, with many interesting open problems. For example, a theorem of Breuer, Guralnick and Kantor (2008) implies that the generating graph of every non-abelian finite simple group is connected with diameter 2 (the proof uses probabilistic and computational methods, based on fixed point ratio estimates). A related theorem of Breuer, Guralnick, Lucchini, Maróti and Nagy (2010) states that $\Gamma(G)$ has a Hamiltonian cycle for all sufficiently large finite simple groups $G$, and they make the following conjecture:

Conjecture. Let $G$ be a finite simple group. Then $\Gamma(G)$ contains a Hamiltonian cycle.
In fact, a much more general conjecture is proposed:
Conjecture. Let $G$ be a finite group with $|G| \geqslant 4$. Then $\Gamma(G)$ contains a Hamiltonian cycle if and only if $G/N$ is cyclic for all non-trivial normal subgroups $N$ of $G$.

In joint work with S. Guest, we have shown $\Gamma(G)$ is connected with diameter 2 for any almost simple cyclic extension $G$ of ${\rm PSL}_{n}(q)$, and work on extending this result to all appropriate almost simple groups is in progress:

T.C. Burness and S. Guest
On the uniform spread of almost simple linear groups
Nagoya Mathematical Journal 209 (2013), 35-109

In a slightly different direction, E. Crestani (Università degli Studi di Padova) and I have investigated the diameter of some connected graphs that arise naturally from the generating graphs of 2-generated direct powers of simple groups:

T.C. Burness and E. Crestani
On the generating graph of direct powers of a simple group
Journal of Algebraic Combinatorics 38 (2013), 329-350

### Representation theory

Let $G$ be a group, let $H$ be a subgroup of $G$, let $K$ be a field and let $V$ be an irreducible $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is also irreducible as a $KH$-module (via restriction). Given a group $G$ and a field $K$, a basic problem in representation theory is to determine all the irreducible triples $(G,H,V)$.

This problem has been well-studied in the case where $G$ is a simple algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$ and $H$ is a closed positive-dimensional subgroup. Indeed, irreducible triples arise naturally in the investigation of the maximal closed subgroups of such groups, which has been a major area of research for many years.

In the 1950s, Dynkin handled the case where $p = 0$ and $H$ is connected. Later, this work was extended to fields of positive characteristic by Seitz and Testerman in the 1980s, which was part of a wider study of the subgroup structure of finite and algebraic simple groups. For disconnected (positive-dimensional) subgroups of exceptional groups, the corresponding irreducible triples have been determined by Ghandour (2010), hence current interest focusses on disconnected subgroups of classical algebraic groups.

In joint work with S. Ghandour (Université Libanaise), C. Marion (EPFL) and D. Testerman (EPFL) we study the irreducible triples $(G,H,V)$, where $G = Cl(W)$ is classical and $H$ is closed, disconnected and positive-dimensional (here $Cl(W)$ denotes one of the classical groups ${\rm SL}(W), {\rm Sp}(W)$ or ${\rm SO}(W)$). First, we assume $H$ is a maximal subgroup: by a theorem of Liebeck and Seitz (1998), $H$ is either geometric (that is, $H$ is defined in terms of the underlying geometry of the natural module $W$) or non-geometric (and almost simple), and we use completely different methods in each case. The corresponding irreducible triples are determined in the following papers:

T.C. Burness, S. Ghandour and D.M. Testerman
Irreducible geometric subgroups of classical algebraic groups
Memoirs of the American Mathematical Society 239 (2016), no. 1130, vi+88 pages ISBN: 978-1-4704-1494-8
T.C. Burness, S. Ghandour, C. Marion and D.M. Testerman
Irreducible almost simple subgroups of classical algebraic groups
Memoirs of the American Mathematical Society 236 (2015), no. 1114, v+110 pages ISBN: 978-1-4704-1046-9

The next step is to extend our results to all disconnected positive-dimensional subgroups, using induction and our earlier work on maximal subgroups. The main theorem of the following paper essentially achieves this goal: either $(G,H,V)$ is known, or $G$ is an orthogonal group, $V$ is a spin module and $H$ preserves an orthogonal decomposition of the natural $KG$-module:

T.C. Burness, C. Marion and D.M. Testerman
On irreducible subgroups of simple algebraic groups
Mathematische Annalen, to appear