arXiv subjects

Timothy C. Burness

Publications and source records attributed to Timothy C. Burness.

At least 37 records · Page 2Linked to original sources

arXiv2022

On the Saxl graphs of primitive groups with soluble stabilisers

Let $G$ be a transitive permutation group on a finite set $Ω$ and recall that a base for $G$ is a subset of $Ω$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$ then we can study the Saxl graph $Σ(G)$ of $G$, which has vertex set $Ω$ and two vertices are adjacent if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most $2$ when $G$ is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of $Σ(G)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $Σ(G)$.

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arXiv2022

On the classification of extremely primitive affine groups

Let $G$ be a finite non-regular primitive permutation group on a set $Ω$ with point stabiliser $G_α$. Then $G$ is said to be extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collection of groups. Moreover, if one assumes Wall's conjecture on the number of maximal subgroups of an almost simple group, then there is an explicit list of candidates, each of which has been eliminated in a recent paper by Burness and Thomas. So, modulo Wall's conjecture, the classification of extremely primitive groups is complete. In this paper we adopt a different approach, which allows us to complete this classification in full generality, independent of the veracity or otherwise of Wall's conjecture in the almost simple setting. Our method relies on recent work of Fawcett, Lee and others on the existence of regular orbits of almost simple groups acting on irreducible modules.

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arXiv2021

Almost elusive permutation groups

Let $G$ be a nontrivial transitive permutation group on a finite set $Ω$. An element of $G$ is said to be a derangement if it has no fixed points on $Ω$. From the orbit counting lemma, it follows that $G$ contains a derangement, and in fact $G$ contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that $G$ is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or $2$-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.

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arXiv2021

Finite groups, minimal bases and the intersection number

Let $G$ be a finite group and recall that the Frattini subgroup ${\rm Frat}(G)$ is the intersection of all the maximal subgroups of $G$. In this paper, we investigate the intersection number of $G$, denoted $α(G)$, which is the minimal number of maximal subgroups whose intersection coincides with ${\rm Frat}(G)$. In earlier work, we studied $α(G)$ in the special case where $G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that $α(G) \leqslant 4$ for every almost simple group $G$, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups $G$ we present best possible bounds on a related invariant $β(G)$, which we call the base number of $G$. In this setting, $β(G)$ is the minimal base size of $G$ as we range over all faithful primitive actions of the group and we prove that the bound $β(G) \leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group $S_{ab}$ on the set of partitions of $[1,ab]$ into $a$ parts of size $b$, determining the exact base size for $a \geqslant b$. This extends earlier work of Benbenishty, Cohen and Niemeyer.

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arXiv2021

On soluble subgroups of sporadic groups

Let $G$ be an almost simple sporadic group and let $H$ be a soluble subgroup of $G$. In this paper we prove that there exists $x,y \in G$ such that $H \cap H^x \cap H^y=1$, which is equivalent to the bound $b(G,H) \leqslant 3$ with respect to the base size of $G$ on the set of cosets of $H$. This bound is best possible. In this setting, our main result establishes a strong form of a more general conjecture of Vdovin on the intersection of conjugate soluble subgroups of finite groups. The proof uses a combination of computational and probabilistic methods.

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arXiv2021

Permutation groups with restricted stabilizers

Fix a positive integer $d$ and let $Γ_d$ be the class of finite groups without sections isomorphic to the alternating group $A_d$. The groups in $Γ_d$ were studied by Babai, Cameron and Pálfy in the 1980s and they determined bounds on the order of a primitive permutation group with this property, which have found a wide range of applications. Subsequently, results on the base sizes of such groups were also obtained. In this paper we replace the structural conditions on the group by restrictions on its point stabilizers, and we obtain similar, and sometimes stronger conclusions. For example, we prove that there is a linear function $f$ such that the base size of any finite primitive group with point stabilizers in $Γ_d$ is at most $f(d)$. This generalizes a recent result of the first author on primitive groups with solvable point stabilizers. For non-affine primitive groups we obtain stronger results, assuming only that stabilizers of $c$ points lie in $Γ_d$. We also show that if $G$ is any permutation group of degree $n$ whose $c$-point stabilizers lie in $Γ_d$, then $|G| \leqslant ((1+o_c(1))d/e)^{n-1}$. This asymptotically extends and improves a $d^{n-1}$ upper bound on $|G|$ obtained by Babai, Cameron and Pálfy assuming $G \in Γ_d$.

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arXiv2021

On the involution fixity of simple groups

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

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arXiv2021

Base sizes for primitive groups with soluble stabilisers

Let $G$ be a finite primitive permutation group on a set $Ω$ with point stabiliser $H$. Recall that a subset of $Ω$ is a base for $G$ if its pointwise stabiliser is trivial. We define the base size of $G$, denoted $b(G,H)$, to be the minimal size of a base for $G$. Determining the base size of a group is a fundamental problem in permutation group theory, with a long history stretching back to the 19th century. Here one of our main motivations is a theorem of Seress from 1996, which states that $b(G,H) \leqslant 4$ if $G$ is soluble. In this paper we extend Seress' result by proving that $b(G,H) \leqslant 5$ for all finite primitive groups $G$ with a soluble point stabiliser $H$. This bound is best possible. We also determine the exact base size for all almost simple groups and we study random bases in this setting. For example, we prove that the probability that $4$ random elements in $Ω$ form a base tends to $1$ as $|G|$ tends to infinity.

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arXiv2021

The spread of a finite group

A group $G$ is said to be $\frac{3}{2}$-generated if every nontrivial element belongs to a generating pair. It is easy to see that if $G$ has this property then every proper quotient of $G$ is cyclic. In this paper we prove that the converse is true for finite groups, which settles a conjecture of Breuer, Guralnick and Kantor from 2008. In fact, we prove a much stronger result, which solves a problem posed by Brenner and Wiegold in 1975. Namely, if $G$ is a finite group and every proper quotient of $G$ is cyclic, then for any pair of nontrivial elements $x_1,x_2 \in G$, there exists $y \in G$ such that $G = \langle x_1, y \rangle = \langle x_2, y \rangle$. In other words, $s(G) \geqslant 2$, where $s(G)$ is the spread of $G$. Moreover, if $u(G)$ denotes the more restrictive uniform spread of $G$, then we can completely characterise the finite groups $G$ with $u(G) = 0$ and $u(G)=1$. To prove these results, we first establish a reduction to almost simple groups. For simple groups, the result was proved by Guralnick and Kantor in 2000 using probabilistic methods and since then the almost simple groups have been the subject of several papers. By combining our reduction theorem and this earlier work, it remains to handle the groups whose socles are exceptional groups of Lie type and this is the case we treat in this paper.

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arXiv2020

The classification of extremely primitive groups

Let $G$ be a finite primitive permutation group on a set $Ω$ with nontrivial point stabilizer $G_α$. We say that $G$ is extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$. These groups arise naturally in several different contexts and their study can be traced back to work of Manning in the 1920s. In this paper, we determine the almost simple extremely primitive groups with socle an exceptional group of Lie type. By combining this result with earlier work of Burness, Praeger and Seress, this completes the classification of the almost simple extremely primitive groups. Moreover, in view of results by Mann, Praeger and Seress, our main theorem gives a complete classification of all finite extremely primitive groups, up to finitely many affine exceptions (and it is conjectured that there are no exceptions). Along the way, we also establish several new results on base sizes for primitive actions of exceptional groups, which may be of independent interest.

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arXiv2020

On solvable factors of almost simple groups

Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Li and Xia describe the factorizations of almost simple groups with a solvable factor $H$. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is identified. Building on this earlier work, in this paper we compute a sharp lower bound on the order of a solvable factor of every almost simple group and we determine the exact factorizations with a solvable factor. As an application, we describe the finite primitive permutation groups with a nilpotent regular subgroup, extending classical results of Burnside and Schur on cyclic regular subgroups, and more recent work of Li in the abelian case.

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arXiv2020

A note on extremely primitive affine groups

Let $G$ be a finite primitive permutation group on a set $Ω$ with nontrivial point stabilizer $G_α$. We say that $G$ is extremely primitive if $G_α$ acts primitively on each of its orbits in $Ω\setminus \{α\}$. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.

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arXiv2020

Topological generation of exceptional algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i \in C_i$ such that $\langle g_1, \ldots, g_t \rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $Ω$ is an irreducible subvariety of $G^t$, then the set of tuples in $Ω$ generating a dense subgroup of $G$ is either empty or dense in $Ω$. In the special case $Ω= C_1 \times \cdots \times C_t$, by considering the dimensions of fixed point spaces, we prove that this set is dense when $G$ is an exceptional algebraic group and $t \geqslant 5$, assuming $k$ is not algebraic over a finite field. In fact, for $G=G_2$ we only need $t \geqslant 4$ and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case $t=2$, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random $(r,s)$-generation of exceptional groups.

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arXiv2019

On the minimal dimension of a finite simple group (with an appendix by T.C. Burness and R.M. Guralnick)

Let $G$ be a finite group and let $\mathcal{M}$ be a set of maximal subgroups of $G$. We say that $\mathcal{M}$ is irredundant if the intersection of the subgroups in $\mathcal{M}$ is not equal to the intersection of any proper subset. The minimal dimension of $G$, denoted ${\rm Mindim}(G)$, is the minimal size of a maximal irredundant set of maximal subgroups of $G$. This invariant was recently introduced by Garonzi and Lucchini and they computed the minimal dimension of the alternating groups. In this paper, we prove that ${\rm Mindim}(G) \leqslant 3$ for all finite simple groups, which is best possible, and we compute the exact value for all non-classical simple groups. We also introduce and study two closely related invariants denoted by $α(G)$ and $β(G)$. Here $α(G)$ (respectively $β(G)$) is the minimal size of a set of maximal subgroups (respectively, conjugate maximal subgroups) of $G$ whose intersection coincides with the Frattini subgroup of $G.$ Evidently, ${\rm Mindim}(G) \leqslant α(G) \leqslant β(G)$. For a simple group $G$ we show that $β(G) \leqslant 4$ and $β(G) - α(G) \leqslant 1$, and both upper bounds are best possible.

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arXiv2019

Finite groups, 2-generation and the uniform domination number

Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G = \langle x_i, y\rangle$ for all $i$. The more restrictive notion of uniform spread, denoted $u(G)$, requires $y$ to be chosen from a fixed conjugacy class of $G$, and a theorem of Breuer, Guralnick and Kantor states that $u(G) \geqslant 2$ for every non-abelian finite simple group $G$. For any group with $u(G) \geqslant 1$, we define the uniform domination number $γ_u(G)$ of $G$ to be the minimal size of a subset $S$ of conjugate elements such that for each nontrivial $x \in G$ there exists $y \in S$ with $G = \langle x, y \rangle$ (in this situation, we say that $S$ is a uniform dominating set for $G$). We introduced the latter notion in a recent paper, where we used probabilistic methods to determine close to best possible bounds on $γ_u(G)$ for all simple groups $G$. In this paper we establish several new results on the spread, uniform spread and uniform domination number of finite groups and finite simple groups. For example, we make substantial progress towards a classification of the simple groups $G$ with $γ_u(G)=2$, and we study the associated probability that two randomly chosen conjugate elements form a uniform dominating set for $G$. We also establish new results concerning the $2$-generation of soluble and symmetric groups, and we present several open problems.

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arXiv2019

On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)

An unrefinable chain of a finite group $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth $3$. Here we go much further by determining the finite groups of depth $3$ and $4$. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most $9$, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number-theoretic result of Heath-Brown, which implies that there are infinitely many non-abelian simple groups of length at most $9$. Finally we study the chain difference of $G$ (namely the length minus the depth). We obtain results on groups with chain difference $1$ and $2$, including a complete classification of the simple groups with chain difference $2$, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of $G/R(G)$ in terms of the chain difference of $G$, where $R(G)$ is the soluble radical of $G$.

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arXiv2019

$A_1$-type subgroups containing regular unipotent elements

Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X = {\rm PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_1$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p) = (E_6,13)$ or $(E_7,19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the embedding of $X$ in $G$. We discuss applications of our main result to the study of the subgroup structure of finite groups of Lie type.

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arXiv2019

The length and depth of compact Lie groups

Let $G$ be a connected Lie 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 $G_{i-1}$. In this paper, we introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups $G$. We obtain best possible bounds on the length of $G$ in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on $\dim G'$ in terms of the chain difference of $G$, which is its length minus its depth.

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