arXiv subjects

Timothy C. Burness

Publications and source records attributed to Timothy C. Burness.

54 records · Page 3Linked to original sources

arXiv2018

Irreducible subgroups of simple algebraic groups - a survey

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module. We say that $(G,H,V)$ is an irreducible triple if $V$ is irreducible as a $KH$-module. Determining these triples is a fundamental problem in the representation theory of algebraic groups, which arises naturally in the study of the subgroup structure of classical groups. In the 1980s, Seitz and Testerman extended earlier work of Dynkin on connected subgroups in characteristic zero to all algebraically closed fields. In this article we will survey recent advances towards a classification of irreducible triples for all positive dimensional subgroups of simple algebraic groups.

math.GR↗

arXiv2018

On base sizes for almost simple primitive groups

Let $G \leqslant {\rm Sym}(Ω)$ be a finite almost simple primitive permutation group, with socle $G_0$ and point stabilizer $H$. A subset of $Ω$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$, denoted $b(G)$, is the minimal size of a base. We say that $G$ is standard if $G_0 = A_n$ and $Ω$ is an orbit of subsets or partitions of $\{1, \ldots, n\}$, or if $G_0$ is a classical group and $Ω$ is an orbit of subspaces (or pairs of subspaces) of the natural module for $G_0$. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have $b(G) \leqslant 7$ for every non-standard group $G$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action on $24$ points. In this paper, we extend this result by classifying the non-standard groups with $b(G)=6$. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

math.GR↗

arXiv2018

On the Saxl graph of a permutation group

Let $G$ be a permutation group on a set $Ω$. A subset of $Ω$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $Σ(G)$, which we call the Saxl graph of $G$. The vertices of $Σ(G)$ are the points of $Ω$, and two vertices are adjacent if they form a base for $G$. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of $Σ(G)$ for a finite transitive group $G$, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if $G$ is a primitive group with a base of size $2$, then the diameter of $Σ(G)$ is at most $2$. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabiliser of $G$ is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter.

math.GR↗

arXiv2018

The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.

math.GR↗

arXiv2018

On the uniform domination number of a finite simple group

Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$ contains a conjugacy class $C$ such that for each non-identity element $x \in G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep result, we introduce a new invariant $γ_u(G)$, which we call the uniform domination number of $G$. This is the minimal size of a subset $S$ of conjugate elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G = \langle x, s \rangle$. (This invariant is closely related to the total domination number of the generating graph of $G$, which explains our choice of terminology.) By the result of Guralnick and Kantor, we have $γ_u(G) \leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper is to determine close to best possible bounds on $γ_u(G)$ for each family of simple groups. For example, we will prove that there are infinitely many non-abelian simple groups $G$ with $γ_u(G) = 2$. To do this, we develop a probabilistic approach, based on fixed point ratio estimates. We also establish a connection to the theory of bases for permutation groups, which allows us to apply recent results on base sizes for primitive actions of simple groups.

math.GR↗

arXiv2018

On the involution fixity of exceptional groups of Lie type

The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree $n$ is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if $T$ is the socle of such a group, then either ${\rm ifix}(T) > n^{1/3}$, or ${\rm ifix}(T) = 1$ and $T = {}^2B_2(q)$ is a Suzuki group in its natural $2$-transitive action of degree $n=q^2+1$. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with ${\rm ifix}(T) \leqslant n^{4/9}$. This extends recent work of Liebeck and Shalev, who established the bound ${\rm ifix}(T) > n^{1/6}$ for every almost simple primitive group of degree $n$ with socle $T$ (with a prescribed list of exceptions). Finally, by combining our results with the Lang-Weil estimates from algebraic geometry, we determine bounds on a natural analogue of involution fixity for primitive actions of exceptional algebraic groups over algebraically closed fields.

math.GR↗

arXiv2017

The depth of a finite simple group

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.

math.GR↗

arXiv2017

Simple groups, generation and probabilistic methods

It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, $(a,b)$-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

math.GR↗

arXiv2017

Simple groups, fixed point ratios and applications

The study of fixed point ratios is a classical topic in permutation group theory, with a long history stretching back to the origins of the subject in the 19th century. Fixed point ratios arise naturally in many different contexts, finding a wide range of applications. In this survey article we focus on fixed point ratios for simple groups of Lie type, highlighting some of the main results, applications and related problems.

math.GR↗

arXiv2017

Permutation groups and derangements of odd prime order

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency.

math.GR↗

arXiv2017

Locally elusive classical groups

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the $r$-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of $r$-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where $H$ is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.

math.GR↗

arXiv2016

Generation of second maximal subgroups and the existence of special primes

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next level of the subgroup lattice - the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^r-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

math.GR↗

arXiv2016

On irreducible subgroups of simple algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case where $G$ is an orthogonal group, $V$ is a spin module and $H$ normalizes an orthogonal decomposition of the natural $KG$-module.

math.GR↗

arXiv2015

Primitive permutation groups and derangements of prime power order

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs.

math.GR↗

arXiv2014

On the prime graph of simple groups

Let $G$ be a finite group, let $π(G)$ be the set of prime divisors of $|G|$ and let $Γ(G)$ be the prime graph of $G$. This graph has vertex set $π(G)$, and two vertices $r$ and $s$ are adjacent if and only if $G$ contains an element of order $rs$. Many properties of these graphs have been studied in recent years, with a particular focus on the prime graphs of finite simple groups. In this note, we determine the pairs $(G,H)$, where $G$ is simple and $H$ is a proper subgroup of $G$ such that $Γ(G) = Γ(H)$.

math.GR↗

arXiv2014

Large subgroups of simple groups

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an analogous result for simple algebraic groups (in this context, largeness is defined in terms of dimension). An application to triple factorisations of simple groups (both finite and algebraic) is discussed.

math.GR↗

arXiv2014

Derangements in primitive permutation groups, with an application to character theory

Let $G$ be a finite primitive permutation group and let $κ(G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $κ(G) \geqslant 1$. In this paper we classify the groups $G$ with $κ(G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $κ(G)=2$, including a complete classification for almost simple groups.

math.GR↗

arXiv2013

Irreducible geometric subgroups of classical algebraic groups

Let $G$ be a simple classical algebraic group over an algebraically closed field $K$ of characteristic $p \ge 0$ with natural module $W$. Let $H$ be a closed subgroup of $G$ and let $V$ be a non-trivial irreducible tensor-indecomposable $p$-restricted rational $KG$-module such that the restriction of $V$ to $H$ is irreducible. In this paper we classify all such triples $(G,H,V)$, where $H$ is a maximal closed disconnected positive-dimensional subgroup of $G$, and $H$ preserves a natural geometric structure on $W$.

math.GR↗