Searcharxiv⌕ Search

arXiv subjects

Gareth Tracey

Publications and source records attributed to Gareth Tracey.

At least 19 recordsLinked to original sources
arXiv2026

Forcible groups and Frattini covers

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 $G$ forces $H$. Complementing a classical result of Thompson--Mann, and answering a recent question of the third author, we show that a finite group is forcible if and only if it is abelian and its Sylow subgroups are elementary-by-cyclic. We also prove relative forcibility results for abelian $p$-groups in the settings of powerful $p$-groups and $p$-groups of bounded nilpotency class.

math.GR↗
arXiv2026

Diameter bounds for arbitrary finite groups and applications

We prove a strong general-purpose bound for the diameter of a finite group depending only on the diameters of its composition factors and the maximal exponent of a normal abelian section. There are a number of notable applications: (1) if $G$ is a finite soluble group of exponent $e$, $\mathrm{diam}(G) \ll e (\log |G|)^8$, (2) anabelian groups with bounded-rank composition factors have polylogarithmic diameter, (3) transitive soluble subgroups of $S_n$ have diameter $\ll n^5$, and (4) Grigorchuk's gap conjecture holds for any finitely generated group acting faithfully on a bounded-degree rooted tree. Additionally, conditional on Babai's conjecture, (5) any transitive permutation group of degree $n$ has diameter bounded by a polynomial in $n$ (a folkloric conjecture), and (6) Grigorchuk's gap conjecture holds for residually finite groups, and thus the conjecture reduces to the simple case.

math.GR↗
arXiv2025

On finite groups with soluble centralisers

We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element centralisers are all soluble, and a reduction theorem for the case in which all non-central $\pi$-elements have soluble centralisers, for a suitable collection $\pi$ of primes. Our results yield further descriptions under mild local conditions and have applications to groups with soluble involution centralisers, as well as to questions concerning non-commuting graphs.

math.GR↗
arXiv2025

Groups with conjugacy classes of coprime sizes

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and $y$ are $\pi$-regular elements for some set of primes $\pi$, then $x^G y^G$ is a $\pi$-regular conjugacy class in $G$. The latter statement was previously known for $\pi$-separable groups $G$ and this generalisation permits us to extend several results concerning the common divisor graph on $p$-regular conjugacy classes, for some prime $p$.

math.GR↗
arXiv2025

Subgroups of symmetric groups: enumeration and asymptotic properties

In this paper, we prove that the symmetric group $\mathrm{S}_n$ has $2^{n^2/16+o(n^2)}$ subgroups, settling a conjecture of Pyber from 1993. We also derive asymptotically sharp upper and lower bounds on the number of subgroups of $\mathrm{S}_n$ of various kinds, including the number of $p$-subgroups. In addition, we prove a range of theorems about random subgroups of $\mathrm{S}_n$. In particular, we prove the surprising result that for infinitely many $n$, the probability that a random subgroup of $\mathrm{S}_n$ is nilpotent is bounded away from $1$.

math.GR↗
arXiv2024

The Chebotarev invariant for direct products of nonabelian finite simple groups

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ invariably generates $G$ if $\{g_1^{x_1}, \ldots , g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected value of the random variable $n$ that is minimal subject to the requirement that $n$ randomly chosen elements of $G$ invariably generate $G$. In this paper, we show that if $G$ is a nonabelian finite simple group, then $C(G)$ is absolutely bounded. More generally, we show that if $G$ is a direct product of $k$ nonabelian finite simple groups, then $C(G)=\log{k}/\log{\alpha(G)}+O(1)$, where $\alpha$ is an invariant completely determined by the proportion of derangements of the primitive permutation actions of the factors in $G$. It follows from the proof of the Boston-Shalev conjecture that $C(G)=O(\log{k})$. We also derive sharp bounds on the expected number of generators for $G$.

math.GR↗
arXiv2024

Minimal sized generating sets of permutation groups

We present a randomised variant of an algorithm of Lucchini and Thakkar for finding a smallest sized generating set in a finite group, which has polynomial time expected running time in finite permutation groups.

math.GR↗
arXiv2024

Weakly subnormal subgroups and variations of the Baer-Suzuki theorem

A subgroup $R$ of a finite group $G$ is weakly subnormal in $G$ if $R$ is not subnormal in $G$ but it is subnormal in every proper overgroup of $R$ in $G$. In this paper, we first classify all finite groups $G$ which contains a weakly subnormal $p$-subgroup for some prime $p$. We then determine all finite groups containing a cyclic weakly subnormal $p$-subgroup. As applications, we prove a number of variations of the Baer-Suzuki theorem using the orders of certain group elements.

math.GR↗
arXiv2023

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

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 $H$ is always closed for the pro-nilpotent topology and compute its closure for the pro-$\mathbf{G}_p$ and pro-$\mathbf{V}_p$ topologies, where $\mathbf{G}_p$ and $\mathbf{V}_p$ denote respectively the pseudovariety of finite $p$-groups and the pseudovariety of finite groups having a normal Sylow $p$-subgroup with quotient an abelian group of exponent dividing $p-1$. More generally, given any nonempty set $P$ of primes, we consider the pseudovariety $\mathbf{G}_P$ of all finite groups having order a product of primes in $P$.

math.GR↗
arXiv2023

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

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? is the closure of $H$ in $F$ for the pro-(met)abelian topology finitely generated? We show also that if the latter question has a positive answer, then we can effectively construct a basis for the closure, and the closure has decidable membership problem in any case. Moreover, it is decidable whether $H$ is closed for the pro-${\bf V}$ topology when ${\bf V}$ is an equational pseudovariety of finite groups, such as the pseudovariety ${\bf S}_k$ of all finite solvable groups with derived length $\leq k$. We also connect the pro-abelian topology with the topologies defined by abelian groups of bounded exponent.

math.GR↗
arXiv2023

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

Let $F$ be a free group of arbitrary rank and let $H$ be a finitely generated subgroup of $F$. Given a pseudovariety $\mathbf{V}$ of finite groups, i.e. a class of finite groups closed under taking subgroups, quotients and finitary direct products, we endow $F$ with its pro-$\mathbf{V}$ topology. Our main result states that it is decidable whether $H$ is $\mathbf{Su}$-dense, where $\mathbf{Su}\subset \mathbf{S}$ denote respectively the pseudovarieties of all finite supersolvable groups and all finite solvable groups. Our motivation stems from the following open problem: is it decidable whether $H$ is $\mathbf{S}$-dense?

math.GR↗
arXiv2023

On the pseudovariety of groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$

We introduce the pseudovariety of finite groups $\mathbf{U} = \displaystyle\bigvee_{p \in \mathbb{P}} {\bf Ab}(p) \ast {\bf Ab}(p-1)$, where $\mathbb{P}$ is the set of all primes. We show that $\mathbf{U}$ consists of all finite supersolvable groups with elementary abelian derived subgroup and abelian Sylow subgroups, being therefore decidable. We prove that it is decidable whether or not a finitely generated subgroup of a free group is closed or dense for the pro-${\bf U}$ topology. We consider also the pseudovariety of finite groups ${\bf Ab}(p) \ast {\bf Ab}(d)$ (where $p$ is a prime and $d$ divides $p-1$). We study the pro-$({\bf Ab}(p) \ast {\bf Ab}(d))$ topology on a free group and construct the unique generator of minimum size of the pseudovariety ${\bf Ab}(p) \ast {\bf Ab}(d)$. Finally, we prove that the variety of groups generated by ${\bf U}$ is the variety of all metabelian groups, obtaining also results on the varieties generated by a Baumslag-Solitar group of the form $BS(1,q)$ for $q$ prime.

math.GR↗
arXiv2021

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

A group $G$ is said to be totally $2$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits for the induced action on $\Omega\times \Omega$. We prove that there are precisely $47$ finite totally $2$-closed groups with trivial Fitting subgroup. Each of these groups is a direct product of pairwise non-isomorphic sporadic simple groups, with the direct factors coming from the Janko groups $\mathrm{J}_1, \mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and the Monster $\mathbb{M}$. These are the first known examples of insoluble totally $2$-closed groups. As a by-product of our methods, we develop several tools for studying $2$-closures of transitive permutation groups -- a vital tool in the study of representations of finite groups as automorphism groups of digraphs. We also prove a dual to a 1939 theorem of Frucht from Algebraic Graph Theory.

math.GR↗
arXiv2021

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

In 1878, Jordan proved that if a finite group $G$ has a faithful representation of dimension $n$ over $\mathbb{C}$, then $G$ has a normal abelian subgroup with index bounded above by a function of $n$. The same result fails if one replaces $\mathbb{C}$ by a field of positive characteristic, due to the presence of large unipotent and/or Lie type subgroups. For this reason, a long-standing problem in group and representation theory has been to find the "correct analogue" of Jordan's theorem in characteristic $p>0$. Progress has been made in a number of different directions, most notably by Brauer and Feit in 1966; by Collins in 2008; and by Larsen and Pink in 2011. With a 1968 theorem of Steinberg in mind (which shows that a significant proportion of elements in a simple group of Lie type are unipotent), we prove in this paper that if a finite group $G$ has a faithful representation over a field of characteristic $p$, then a significant proportion of the elements of $G$ must have $p$-power order. We prove similar results for permutation groups, and present a general method for counting $p$-elements in finite groups. All of our results are best possible.

math.GR↗
arXiv2021

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

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 $\lfloor cn/\sqrt{\log{n}}\rfloor$ elements, where $c:=\sqrt{3}/2$. Owing to the transitive group $D_8\circ D_8$ of degree $8$, this upper bound is best possible. Our new result improves a 2018 paper by the author, and makes use of the recent classification of transitive groups of degree $48$.

math.GR↗
arXiv2021

The transitive groups of degree 48 and some applications

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195\,826\,352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive graphs of degree $48$, of which there are $1\,538\,868\,366$, all but $0.1625\%$ of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published by the third author as a separate paper

math.GR↗