Gareth Tracey

activity

20152026

most citedOn the generalized Fitting height and insoluble length of finite groups

6 citations · 17 across the 17 of their papers we have counts for

collaborators

25 papers

math.GR2026

Forcible groups and Frattini covers

Sean Eberhard, Robert M. Guralnick, Carl Schildkraut +1

We say that a finite group $G$ *forces* a finite group $H$ if every finite cover of $G$ contains a subgroup isomorphic to $H$ , and we say $H$ is *forcible* if some finite group $G$ …

math.GR2024

The Chebotarev invariant for direct products of nonabelian finite simple groups

Jessica Anzanello, Andrea Lucchini, Gareth Tracey

A subset ${g_{1}, \dots, g_{d}}$ of a finite group $G$ invariably generates $G$ if ${g_{1}^{x_{1}}, \dots, g_{d}^{x_{d}}}$ generates $G$ for every choice of $x_{i} \in G$ . The Chebotarev…

math.GR2023

The pro-supersolvable topology on a free group: deciding denseness

Claude Marion, Pedro V. Silva, Gareth Tracey

Let $F$ be a free group of arbitrary rank and let $H$ be a finitely generated subgroup of $F$ . Given a pseudovariety $V$ of finite groups, i.e. a class of finite groups cl…

math.GR2023

On the closure of cyclic subgroups of a free group in pro-V topologies

Claude Marion, Pedro V. Silva, Gareth Tracey

We determine the closure of a cyclic subgroup $H$ of a free group for the pro-{\bf V} topology when {\bf V} is an extension-closed pseudovariety of finite groups. We show that $H$ …

math.GR2023★ 1 cited

The pro- $k$ -solvable topology on a free group

Claude Marion, Pedro V. Silva, Gareth Tracey

We prove that, given a finitely generated subgroup $H$ of a free group $F$ , the following questions are decidable: is $H$ closed (dense) in $F$ for the pro-(met)abelian topology? i…

math.GR2023

On the pseudovariety of groups $U = p \in P ⋁ Ab (p) * Ab (p - 1)$

Claude Marion, Pedro V. Silva, Gareth Tracey

We introduce the pseudovariety of finite groups $U = p \in P ⋁ Ab (p) * Ab (p - 1)$ , where $P$ is the set of all primes.…

math.GR2021

Comparing the order and the minimal number of generators of a finite irreducible linear group

Derek Holt, Gareth Tracey

We prove that $d (G) lo g ∣ G ∣ = O (n^{2} lo g q)$ for irreducible subgroups $G$ of GL $(n, q)$ , and estimate the associated constants. The result is motivated by attempts to bound the co…

math.GR2021★ 3 cited

Totally $2$ -closed finite groups with trivial Fitting subgroup

Majid Arezoomand, Mohammad A. Iranmanesh, Cheryl E. Praeger +1

A group $G$ is said to be totally $2$ -closed if in each of its faithful permutation representations, say on a set $Ω$ , $G$ is the largest subgroup of $Sym (Ω)$ which leaves…

math.GR2021★ 1 cited

On the proportion of $p$ -elements in a finite group, and a modular Jordan type theorem

Gareth Tracey

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $C$ , then $G$ has a normal abelian subgroup with index bounded above…

math.GR2021

The transitive groups of degree 48 and some applications

Derek Holt, Gordon Royle, Gareth Tracey

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195826352$ conjugacy classes of transitive subgroups of the sym…

math.GR2021★ 3 cited

Sharp upper bounds on the minimal number of elements required to generate a transitive permutation group

Gareth Tracey

The purpose of this paper is to prove that if $G$ is a transitive permutation group of degree $n \geq 2$ , then $G$ can be generated by $⌊ c n / lo g n ⌋$ elements, w…

math.GR2021

Some simplifications in the proof of the Sims conjecture

László Pyber, Gareth Tracey

We prove an elementary lemma concerning primitive amalgams and use it to greatly simplify the proof of the Sims conjecture in the case of almost simple groups.

math.GR2020

Locally finite $p$ -groups with a left 3-Engel element whose normal closure is not nilpotent

Anastasia Hadjievangelou, Marialaura Noce, Gunnar Traustason

For any odd prime $p$ , we give an example of a locally finite $p$ -group $G$ containing a left 3-Engel element $x$ where $⟨ x ⟩^{G}$ is not nilpotent.

math.GR2020

An upper bound on the Chebotarev invariant of a finite group

Andrea Lucchini, Gareth Tracey

A subset ${g_{1}, \dots, g_{d}}$ of a finite group $G$ invariably generates $G$ if the set ${g_{1}^{x_{1}}, \dots, g_{d}^{x_{d}}}$ generates $G$ for every choice of $x_{i} \in G$ . The Che…

math.GR2020

Left $3$ -Engel elements in groups of exponent $60$

Gareth Tracey, Gunnar Traustason

Let $G$ be a group and let $x \in G$ be a left $3$ -Engel element of order dividing $60$ . Suppose furthermore that $⟨ x ⟩^{G}$ has no elements of order $8$ , $9$ and $25$ .…

math.GR2020★ 1 cited

On finite totally 2-closed groups

Alireza Abdollahi, Majid Arezoomand, Gareth Tracey

An abstract group $G$ is called totally $2$ -closed if $H = H^{(2), Ω}$ for any set $Ω$ with $G ≅ H \leq Sym (Ω)$ , where $H^{(2), Ω}$ is the largest subgroup of $Sym (Ω)$ who…

math.GR2019

Engel elements in weakly branch groups

Gustavo A. Fernández-Alcober, Marialaura Noce, Gareth M. Tracey

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set o…

math.GR2019★ 6 cited

On the generalized Fitting height and insoluble length of finite groups

Robert M. Guralnick, Gareth Tracey

We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a genera…

math.GR2018

A left $3$ -Engel element whose normal closure is not nilpotent

Marialaura Noce, Gareth M. Tracey, Gunnar Traustason

We give an example of a locally nilpotent group $G$ containing a left $3$ -Engel element $x$ where $⟨ x ⟩^{G}$ is not nilpotent.

math.GR2018

Finite groups with large Chebotarev invariant

Andrea Lucchini, Gareth Tracey

A subset ${g_{1}, \dots, g_{d}}$ of a finite group $G$ is said to invariably generate $G$ if the set ${g_{1}^{x_{1}}, \dots, g_{d}^{x_{d}}}$ generates $G$ for every choice of $x_i \in G…

math.GR2018★ 1 cited

Generating maximal subgroups of finite almost simple groups

Andrea Lucchini, Claude Marion, Gareth Tracey

For a finite group $G$ , let $d (G)$ denote the minimal number of elements required to generate $G$ . In this paper, given a finite almost simple group $G$ and any maximal subgroup $H…

math.GR2018

Invariable generation of permutation and linear groups

Gareth M. Tracey

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ …

math.GR2016★ 1 cited

Comparing the order and the minimal number of generators of a transitive permutation group

Gareth M. Tracey

We prove that if $G$ is a transitive permutation group, then $d (G) lo g ∣ G ∣ / n^{2}$ tends to $0$ as $n$ tends to $\infty$ .

math.GR2015

Generating minimally transitive permutation groups

Gareth M. Tracey

We improve the upper bounds (in terms of $n$ ) in [9] and [13] on the minimal number of elements required to generate a minimally transitive permutation group of degree $n$ .

Forcible groups and Frattini covers

The Chebotarev invariant for direct products of nonabelian finite simple groups

The pro-supersolvable topology on a free group: deciding denseness

On the closure of cyclic subgroups of a free group in pro-V topologies

The pro-k-solvable topology on a free group

On the pseudovariety of groups U=p∈P⋁​Ab(p)∗Ab(p−1)

Comparing the order and the minimal number of generators of a finite irreducible linear group

Totally 2-closed finite groups with trivial Fitting subgroup

On the proportion of p-elements in a finite group, and a modular Jordan type theorem

The transitive groups of degree 48 and some applications

Sharp upper bounds on the minimal number of elements required to generate a transitive permutation group

Some simplifications in the proof of the Sims conjecture

Locally finite p-groups with a left 3-Engel element whose normal closure is not nilpotent

An upper bound on the Chebotarev invariant of a finite group

Left 3-Engel elements in groups of exponent 60

On finite totally 2-closed groups

Engel elements in weakly branch groups

On the generalized Fitting height and insoluble length of finite groups

A left 3-Engel element whose normal closure is not nilpotent

Finite groups with large Chebotarev invariant

Generating maximal subgroups of finite almost simple groups

Invariable generation of permutation and linear groups

Comparing the order and the minimal number of generators of a transitive permutation group

Generating minimally transitive permutation groups

The pro- $k$ -solvable topology on a free group

On the pseudovariety of groups $U = p \in P ⋁ Ab (p) * Ab (p - 1)$

Totally $2$ -closed finite groups with trivial Fitting subgroup

On the proportion of $p$ -elements in a finite group, and a modular Jordan type theorem

Locally finite $p$ -groups with a left 3-Engel element whose normal closure is not nilpotent

Left $3$ -Engel elements in groups of exponent $60$

A left $3$ -Engel element whose normal closure is not nilpotent