Timothy C. Burness

activity

20132026

most citedOn the commuting probability of p-elements in a finite group

12 citations · 48 across the 40 of their papers we have counts for

collaborators

57 papers

math.GR2026

On second maximal subgroups of finite groups

Timothy C. Burness, Hangyang Meng

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 su…

math.GR2026

On the regularity of irreducible subgroups of finite classical groups

Timothy C. Burness, Lei Wang

Let $G$ be a finite group and let $τ = (H_{1}, \dots, H_{t})$ be a $t$ -tuple of core-free subgroups of $G$ . We say that $τ$ is regular if $G$ contains elements $g_{1}, \dots, g_{t}$ such…

math.GR2025

On the intersections of nilpotent subgroups in simple groups

Timothy C. Burness, Hong Yi Huang

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…

math.GR2025

On the intersections of Sylow subgroups in almost simple groups

Timothy C. Burness, Hong Yi Huang

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…

math.GR2025

On the depth of subgroups of simple groups

Timothy C. Burness

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 \ma…

math.GR2025★ 1 cited

On fixed-point-free involutions in actions of finite exceptional groups of Lie type

Timothy C. Burness, Mikko Korhonen

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…

math.GR2024

On derangements in simple permutation groups

Timothy C. Burness, Marco Fusari

Let $G ⩽ Sym (Ω)$ 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 o…

math.GR2024

On the regularity number of a finite group and other base-related invariants

Marina Anagnostopoulou-Merkouri, Timothy C. Burness

A $k$ -tuple $(H_{1}, \dots, H_{k})$ 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…

math.GR2023

On the maximal overgroups of Sylow subgroups of finite groups

Barbara Baumeister, Timothy C. Burness, Robert M. Guralnick +1

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…

math.GR2022

Normalisers of maximal tori and a conjecture of Vdovin

Timothy C. Burness, Adam R. Thomas

Let $G = O^{p^{'}} (G ˉ^{F})$ 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 g…

math.GR2022

Strongly base-two groups

Timothy C. Burness, Robert M. Guralnick

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 co…

math.GR2022

On the topological generation of exceptional groups by unipotent elements

Timothy C. Burness

Let $G$ be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $p ⩾ 0$ which is not algebraic over a finite field. Let $\mathc…

math.GR2022

On the generation of simple groups by Sylow subgroups

Timothy C. Burness, Robert M. Guralnick

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 $g \in G$ s…

math.GR2022

On base sizes for primitive groups of product type

Timothy C. Burness, Hong Yi Huang

Let $G ⩽ Sym (Ω)$ 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…

math.GR2021★ 12 cited

On the commuting probability of p-elements in a finite group

Timothy C. Burness, Robert M. Guralnick, Alexander Moretó +1

Let $G$ be a finite group, let $p$ be a prime and let $Pr_{p} (G)$ be the probability that two random $p$ -elements of $G$ commute. In this paper we prove that ${\rm Pr}_p(G) > (…

math.GR2021★ 1 cited

Fixed point ratios for finite primitive groups and applications

Timothy C. Burness, Robert M. Guralnick

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 $fpr (x)$ , is the proportion of points in $…

math.GR2021

On the soluble graph of a finite group

Timothy C. Burness, Andrea Lucchini, Daniele Nemmi

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 comm…

math.GR2021

Topological generation of simple algebraic groups

Timothy C. Burness, Spencer Gerhardt, Robert M. Guralnick

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^{r}$ with $r ⩾ 2$ . In this paper, we consider the genera…

math.GR2021

On the classification of extremely primitive affine groups

Timothy C. Burness, Melissa Lee

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 o…

math.GR2021

On the Saxl graphs of primitive groups with soluble stabilisers

Timothy C. Burness, Hong Yi Huang

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(…

math.GR2021

On soluble subgroups of sporadic groups

Timothy C. Burness

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 i…

math.GR2020

Permutation groups with restricted stabilizers

Timothy C. Burness, Aner Shalev

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, Camero…

math.GR2020

Almost elusive permutation groups

Timothy C. Burness, Emily V. Hall

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 le…

math.GR2020

Finite groups, minimal bases and the intersection number

Timothy C. Burness, Martino Garonzi, Andrea Lucchini

Let $G$ be a finite group and recall that the Frattini subgroup $Frat (G)$ is the intersection of all the maximal subgroups of $G$ . In this paper, we investigate the intersect…