arXiv subjects
Publications and source records attributed to Timothy C. Burness.
A second maximal subgroup of a finite group $G$ is a subgroup $H$ that is maximal in every maximal overgroup of $H$ in $G$. We prove that every maximal subgroup of a prime-index subgroup of a finite group is a second maximal subgroup.
Let $G$ be a finite group and let $\tau = (H_1, \ldots, H_t)$ be a $t$-tuple of core-free subgroups of $G$. We say that $\tau$ is regular if $G$ contains elements $g_1, \ldots, g_t$ such that $\bigcap_i H_i^{g_i} = 1$, which is equivalent to the existence of a regular $G$-orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_t$. Regular tuples were first investigated by Anagnostopoulou-Merkouri and Burness in a paper from 2024, partly motivated by the aim of seeking a natural generalisation of the classical and widely studied concept of a base for a transitive permutation group, which aligns with the special case where the $H_i$ are pairwise conjugate subgroups. In this paper, we focus on the case where $G$ is a finite almost simple classical group and each $H_i$ is a maximal subgroup contained in Aschbacher's collection $\mathcal{S}$ of irreducibly embedded subgroups. Our main theorem determines all the non-regular $t$-tuples of this form with $t \geqslant 2$, which extends earlier work by Burness, Guralnick and Saxl in the base size setting. In particular, we deduce that every pair of maximal subgroups in $\mathcal{S}$ is regular if $n \geqslant 15$, where $n$ is the dimension of the natural module for the socle of $G$, and this lower bound is best possible.
Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set $\{H_p \cap H_p^g \,:\, g \in G\}$ for every prime $p$. For a simple group $G$, in view of a theorem of Mazurov and Zenkov from 1996, the conjecture implies the existence of an element $x \in G$ with $H_p \cap H_p^x = 1$ for all $p$. In turn, this statement implies a conjecture of Vdovin from 2002, which asserts that if $G$ is simple and $H$ is a nilpotent subgroup, then $H \cap H^x = 1$ for some $x \in G$. In this paper, we adopt a probabilistic approach to prove the Lisi-Sabatini conjecture for all non-alternating simple groups. By combining this with earlier work of Kurmazov on nilpotent subgroups of alternating groups, we complete the proof of Vdovin's conjecture. Moreover, by combining our proof with earlier work of Zenkov on alternating groups, we are able to establish a stronger form of Vdovin's conjecture: if $G$ is simple and $A,B$ are nilpotent subgroups, then $A \cap B^x = 1$ for some $x \in G$. To obtain these results, we study the probability that a random pair of Sylow $p$-subgroups in a simple group of Lie type intersect trivially, complementing recent work of Diaconis et al. and Eberhard on symmetric and alternating groups.
The depth of a subgroup $H$ of a finite group $G$ is a positive integer defined with respect to the inclusion of the corresponding complex group algebras $\mathbb{C}H \subseteq \mathbb{C}G$. This notion was originally introduced by Boltje, Danz and Külshammer in 2011, and it has been the subject of numerous papers in recent years. In this paper, we study the depth of core-free subgroups, which allows us to apply powerful computational and probabilistic techniques that were originally designed for studying bases for permutation groups. We use these methods to prove a wide range of new results on the depth of subgroups of almost simple groups, significantly extending the scope of earlier work in this direction. For example, we establish best possible bounds on the depth of irreducible subgroups of classical groups and primitive subgroups of symmetric groups. And with the exception of a handful of open cases involving the Baby Monster, we calculate the exact depth of every subgroup of every almost simple sporadic group. We also present a number of open problems and conjectures.
Let $G$ be a finite almost simple group and let $H$ be a Sylow $p$-subgroup of $G$. As a special case of a theorem of Zenkov, there exist $x,y \in G$ such that $H \cap H^x \cap H^y = 1$. In fact, if $G$ is simple, then a theorem of Mazurov and Zenkov reveals that $H \cap H^x = 1$ for some $x \in G$. However, it is known that the latter property does not extend to all almost simple groups. For example, if $G = S_8$ and $p=2$, then $H \cap H^x \ne 1$ for all $x \in G$. Further work of Zenkov in the 1990s shows that such examples are rare (for instance, there are no such examples if $p \geqslant 5$) and he reduced the classification of all such pairs to the situation where $p=2$ and $G$ is an almost simple group of Lie type defined over a finite field $\mathbb{F}_q$ and either $q=9$ or $q$ is a Mersenne or Fermat prime. In this paper, by adopting a probabilistic approach based on fixed point ratio estimates, we complete Zenkov's classification.
Let $G$ be a nontrivial transitive permutation group on a finite set $Ω$. By a classical theorem of Jordan, $G$ contains a derangement, which is an element with no fixed points on $Ω$. Given a prime divisor $r$ of $|Ω|$, we say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. In a paper from 2011, Burness, Giudici and Wilson essentially reduce the classification of the $r$-elusive primitive groups to the case where $G$ is an almost simple group of Lie type. The classical groups with an $r$-elusive socle have been determined by Burness and Giudici, and in this paper we consider the analogous problem for the exceptional groups of Lie type, focussing on the special case $r=2$. Our main theorem describes all the almost simple primitive exceptional groups with a $2$-elusive socle. In other words, we determine the pairs $(G,M)$, where $G$ is an almost simple exceptional group of Lie type with socle $T$ and $M$ is a core-free maximal subgroup that intersects every conjugacy class of involutions in $T$. Our results are conclusive, with the exception of a finite list of undetermined cases for $T = E_8(q)$, which depend on the existence (or otherwise) of certain almost simple maximal subgroups of $G$ that have not yet been completely classified.
Let $G \leqslant {\rm Sym}(Ω)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $Ω$. Let $Δ(G)$ be the set of derangements in $G$ and define $δ(G) = |Δ(G)|/|G|$ and $Δ(G)^2 = \{ xy \,:\, x,y \in Δ(G)\}$. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that $δ(G) \geqslant 0.016$ and $G = Δ(G)^2$ for all sufficiently large simple transitive groups $G$. In this paper, we extend these results in several directions. For example, we prove that $δ(G) \geqslant 89/325$ and $G = Δ(G)^2$ for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on $δ(G)$ is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.
A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$ is said to be regular if $G$ has a regular orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted $R(G)$, is the smallest positive integer $k$ with the property that every such $k$-tuple is regular. In this paper, we develop some general methods for studying the regularity of subgroup tuples in arbitrary finite groups, and we determine the precise regularity number of all almost simple groups with an alternating or sporadic socle. For example, we prove that $R(S_n) = n-1$ and $R(A_n) = n-2$. We also formulate and investigate natural generalisations of several well-studied problems on base sizes for finite permutation groups, including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend earlier work of Burness, O'Brien and Wilson by proving that $R(G) \leqslant 7$ for every almost simple sporadic group, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$. We also show that every triple of soluble subgroups in an almost simple sporadic group is regular, which generalises recent work of Burness on base sizes for transitive actions of sporadic groups with soluble point stabilisers.
In this paper, we determine the finite groups with a Sylow $r$-subgroup contained in a unique maximal subgroup. The proof involves a reduction to almost simple groups, and our main theorem extends earlier work of Aschbacher in the special case $r=2$. Several applications are presented. This includes some new results on weakly subnormal subgroups of finite groups, which can be used to study variations of the Baer-Suzuki theorem.
Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1, \ldots, x_r) \in X$ such that $\langle x_1, \ldots, x_r \rangle$ is Zariski dense in $G$. We are primarily interested in the case where $X = C_1 \times \cdots \times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising elements of prime order modulo the center of $G$. In this setting, our main theorem gives a complete solution to the problem when $G$ is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives a complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.
Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p \geqslant 0$ which is not algebraic over a finite field. Let $\mathcal{C}_1, \ldots, \mathcal{C}_t$ be non-central conjugacy classes in $G$. In earlier work with Gerhardt and Guralnick, we proved that if $t \geqslant 5$ (or $t \geqslant 4$ if $G = G_2$), then there exist elements $x_i \in \mathcal{C}_i$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense in $G$. Moreover, this bound on $t$ is best possible. Here we establish a more refined version of this result in the special case where $p>0$ and the $\mathcal{C}_i$ are unipotent classes containing elements of order $p$. Indeed, in this setting we completely determine the classes $\mathcal{C}_1, \ldots, \mathcal{C}_t$ for $t \geqslant 2$ such that $\langle x_1, \ldots, x_t \rangle$ is Zariski dense for some $x_i \in \mathcal{C}_i$.
Let $G$ be a finite group, let $H$ be a core-free subgroup and let $b(G,H)$ denote the base size for the action of $G$ on $G/H$. Let $α(G)$ be the number of conjugacy classes of core-free subgroups $H$ of $G$ with $b(G,H) \geqslant 3$. We say that $G$ is a strongly base-two group if $α(G) \leqslant 1$, which means that almost every faithful transitive permutation representation of $G$ has base size $2$. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup.
Let $G = O^{p'}(\bar{G}^F)$ be a finite simple group of Lie type defined over a field of characteristic $p$, where $F$ is a Steinberg endomorphism of the ambient simple algebraic group $\bar{G}$. Let $\bar{T}$ be an $F$-stable maximal torus of $\bar{G}$ and set $N = N_G(\bar{T})$. A conjecture due to Vdovin asserts that if $G \not\cong {\rm L}_3(2)$ then $N \cap N^x$ is a $p$-group for some $x \in G$. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of $G$ on $G/N$, which allows us to prove a stronger, and suitably modified, version of Vdovin's conjecture.
Let $G$ be a finite primitive permutation group on a set $Ω$ and recall that the fixed point ratio of an element $x \in G$, denoted ${\rm fpr}(x)$, is the proportion of points in $Ω$ fixed by $x$. Fixed point ratios in this setting have been studied for many decades, finding a wide range of applications. In this paper, we are interested in comparing ${\rm fpr}(x)$ with the order of $x$. Our main theorem classifies the triples $(G,Ω,x)$ as above with the property that $x$ has prime order $r$ and ${\rm fpr}(x) > 1/(r+1)$. There are several applications. Firstly, we extend earlier work of Guralnick and Magaard by determining the primitive permutation groups of degree $m$ with minimal degree at most $2m/3$. Secondly, our main result plays a key role in recent work of the authors (together with Moretó and Navarro) on the commuting probability of $p$-elements in finite groups. Finally, we use our main theorem to investigate the minimal index of a primitive permutation group, which allows us to answer a question of Bhargava.
Let $G$ be a finite insoluble group with soluble radical $R(G)$. In this paper we investigate the soluble graph of $G$, which is a natural generalisation of the widely studied commuting graph. Here the vertices are the elements in $G \setminus R(G)$, with $x$ adjacent to $y$ if they generate a soluble subgroup of $G$. Our main result states that this graph is always connected and its diameter, denoted $δ_{\mathcal{S}}(G)$, is at most $5$. More precisely, we show that $δ_{\mathcal{S}}(G) \leqslant 3$ if $G$ is not almost simple and we obtain stronger bounds for various families of almost simple groups. For example, we will show that $δ_{\mathcal{S}}(S_n) = 3$ for all $n \geqslant 6$. We also establish the existence of simple groups with $δ_{\mathcal{S}}(G) \geqslant 4$. For instance, we prove that $δ_{\mathcal{S}}(A_{2p+1}) \geqslant 4$ for every Sophie Germain prime $p \geqslant 5$, which demonstrates that our general upper bound of $5$ is close to best possible. We conclude by briefly discussing some variations of the soluble graph construction and we present several open problems.
Let $G \leqslant {\rm Sym}(Ω)$ be a finite permutation group and recall that the base size of $G$ is the minimal size of a subset of $Ω$ with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form $L \wr P \leqslant {\rm Sym}(Ω)$ with soluble point stabilisers, where $Ω= Γ^k$, $L \leqslant {\rm Sym}(Γ)$ and $P \leqslant S_k$ is transitive. This extends recent work of Burness on almost simple primitive groups. We also obtain an expression for the number of regular suborbits of any product type group of the form $L \wr P$ and we classify the groups with a unique regular suborbit under the assumption that $P$ is primitive, which involves extending earlier results due to Seress and Dolfi. We present applications on the Saxl graphs of base-two product type groups and we conclude by establishing several new results on base sizes for general product type primitive groups.
Let $G$ be a finite group, let $p$ be a prime and let ${\rm Pr}_p(G)$ be the probability that two random $p$-elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (p^2+p-1)/p^3$ if and only if $G$ has a normal and abelian Sylow $p$-subgroup, which generalizes previous results on the widely studied commuting probability of a finite group. This bound is best possible in the sense that for each prime $p$ there are groups with ${\rm Pr}_p(G) = (p^2+p-1)/p^3$ and we classify all such groups. Our proof is based on bounding the proportion of $p$-elements in $G$ that commute with a fixed $p$-element in $G \setminus \textbf{O}_p(G)$, which in turn relies on recent work of the first two authors on fixed point ratios for finite primitive permutation groups.
Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this result with recent work of Breuer and Guralnick, we deduce that if $G$ is a finite nonabelian simple group and $r$ is any prime divisor of $|G|$, then $G$ is generated by a Sylow $2$-subgroup and a Sylow $r$-subgroup.