arXiv subjects

Tomasz Popiel

Publications and source records attributed to Tomasz Popiel.

At least 19 recordsLinked to original sources

arXiv2025

Derangements in permutation groups with two orbits

A classical theorem of Jordan asserts that if a group $G$ acts transitively on a finite set of size at least $2$, then $G$ contains a derangement (a fixed-point free element). Generalisations of Jordan's theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper, which says that if $G$ has exactly two orbits and those orbits have equal length $n \geq 2$, then $G$ contains a derangement. We prove this conjecture in the case where $n$ is a product of two primes, and verify it computationally for $n \leq 30$.

math.GR↗

arXiv2025

The maximal subgroups of the Monster

The classification of the maximal subgroups of the Monster $\mathbf{M}$ is a long-standing problem in finite group theory. According to the literature, the classification is complete apart from the question of whether $\mathbf{M}$ contains maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(13)$. However, this conclusion relies on reported claims, with unpublished proofs, that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle $\mathrm{PSL}_2(8)$, $\mathrm{PSL}_2(16)$, or $\mathrm{PSU}_3(4)$. The aim of this paper is to settle all of these questions, and thereby complete the solution to the maximal subgroup problem for $\mathbf{M}$, and for the sporadic simple groups as a whole. Specifically, we prove the existence of two new maximal subgroups of $\mathbf{M}$, isomorphic to the automorphism groups of $\mathrm{PSL}_2(13)$ and $\mathrm{PSU}_3(4)$, and we establish that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSL}_2(8)$ or $\mathrm{PSL}_2(16)$. We also correct the claim that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSU}_3(4)$, and provide evidence that the maximal subgroup $\mathrm{PSL}_2(59)$ (constructed in 2004) does not exist. Our proofs are supported by reproducible computations carried out using the publicly available Python package mmgroup for computing with $\mathbf{M}$ recently developed by M. Seysen. We provide explicit generators for our newly discovered maximal subgroups of $\mathbf{M}$ in mmgroup format.

math.GR↗

arXiv2024

Explicit construction of the maximal subgroups of the Monster

Seysen's Python package mmgroup provides functionality for fast computations within the sporadic simple group $\mathbb{M}$, the Monster. The aim of this work is to present an mmgroup database of maximal subgroups of $\mathbb{M}$: for each conjugacy class $C$ of maximal subgroups in $\mathbb{M}$, we construct explicit group elements in mmgroup and prove that these elements generate a group in $C$. Our generators and the computations verifying correctness are available in accompanying code. The maximal subgroups of $\mathbb{M}$ have been classified in a number of papers spanning several decades; our work constitutes an independent verification of these constructions. We also correct the claim that $\mathbb{M}$ has a maximal subgroup $\mathrm{PSL}_2({59})$, and hence identify a new maximal subgroup $59{:}29$.

math.GR↗

arXiv2024

Recognisability of the sporadic groups by the isomorphism types of their prime graphs

The prime graph of a finite group $G$ is the labelled graph $Γ(G)$ with vertices the prime divisors of $|G|$ and edges the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A group $G$ is recognisable by its prime graph if every group $H$ with $Γ(H)=Γ(G)$ is isomorphic to $G$. Cameron and Maslova have shown that every group that is recognisable by its prime graph is almost simple, which justifies the significant amount of attention that has been given to determining which simple (or almost simple) groups are recognisable by their prime graphs. This problem has been completely solved for certain families of simple groups, including the sporadic groups. A natural extension of the problem is to determine which groups are recognisable by their unlabelled prime graphs, i.e. by the isomorphism types of their prime graphs. There seem to be only very limited results in this direction in the literature. Here we determine which of the sporadic finite simple groups are recognisable by the isomorphism types of their prime graphs. We also show that for every sporadic group $G$ that is not recognisable by the isomorphism type of $Γ(G)$, there are infinitely many groups $H$ with $Γ(H) \cong Γ(G)$.

math.GR↗

arXiv2024

Showcasing straight-line programs with memory via matrix Bruhat decomposition

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalising this idea and, as illustration, construct and analyse straight-line programs for the Bruhat decomposition of $d\times d$ matrices with determinant $1$ over a finite field of order $q$ that have length $O(d^2\log(q))$ and require storing only $O(\log(q))$ matrices during evaluation.

cs.DS↗

arXiv2024

Conjugacy class fusion from four maximal subgroups of the Monster

We determine the conjugacy class fusion from certain maximal subgroups of the Monster to the Monster, to justify the addition of these data to the Character Table Library in the computational algebra system GAP. The maximal subgroups in question are $(\text{PSL}_2(11) {\times} \text{PSL}_2(11)){:}4$, $11^2{:}(5 {\times} 2\text{A}_5)$, $7^2{:}\text{SL}_2(7)$, and $\text{PSL}_2(19){:}2$. Our proofs are supported by reproducible calculations carried out using the Python package mmgroup, a computational construction of the Monster recently developed by Seysen.

math.GR↗

arXiv2023

Indeed, the Monster has no almost simple maximal subgroup with socle $\text{PSL}_2(16)$

The classification of the maximal subgroups of the Monster $\mathbf{M}$ is believed to be complete subject to an unpublished result of Holmes and Wilson asserting that $\mathbf{M}$ has no maximal subgroups that are almost simple with socle isomorphic to $\text{PSL}_2(8)$, $\text{PSL}_2(16)$, or $\text{PSU}_3(4)$. We prove this result for $\text{PSL}_2(16)$, with the intention that the other two cases will be dealt with in an expanded version of this paper. Our proof is supported by reproducible computations carried out using Seysen's publicly available Python package mmgroup for computing with $\mathbf{M}$.

math.GR↗

arXiv2022

$\text{M}$, $\text{B}$ and $\text{Co}_1$ are recognisable by their prime graphs

The prime graph, or Gruenberg--Kegel graph, of a finite group $G$ is the graph $Γ(G)$ whose vertices are the prime divisors of $|G|$, and whose edges are the pairs $\{p,q\}$ for which $G$ contains an element of order $pq$. A finite group $G$ is recognisable by its prime graph if every finite group $H$ with $Γ(H)=Γ(G)$ is isomorphic to $G$. By a result of Cameron and Maslova, every such group must be almost simple, so one natural case to investigate is that in which $G$ is one of the $26$ sporadic simple groups. Existing work of various authors answers the question of recognisability by prime graph for all but three of these groups, namely the Monster, $\text{M}$, the Baby Monster, $\text{B}$, and the first Conway group, $\text{Co}_1$. We prove that these three groups are recognisable by their prime graphs.

math.GR↗

arXiv2021

An exotic presentation of Q_28

We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.

math.AT↗

arXiv2021

Saxl graphs of primitive affine groups with sporadic point stabilisers

Let $G$ be a permutation group on a set $Ω$. A base for $G$ is a subset of $Ω$ whose pointwise stabiliser is trivial, and the base size of $G$ is the minimal cardinality of a base. If $G$ has base size $2$, then the corresponding Saxl graph $Σ(G)$ has vertex set $Ω$ and two vertices are adjacent if they form a base for $G$. A recent conjecture of Burness and Giudici states that if $G$ is a finite primitive permutation group with base size $2$, then $Σ(G)$ has the property that every two vertices have a common neighbour. We investigate this conjecture when $G$ is an affine group and a point stabiliser is an almost quasisimple group whose unique quasisimple subnormal subgroup is a covering group of a sporadic simple group. We verify the conjecture under this assumption, in all but ten cases.

math.GR↗

arXiv2021

Solids in the space of the Veronese surface in even characteristic

We classify the orbits of solids in the projective space $\text{PG}(5,q)$, $q$ even, under the setwise stabiliser $K \cong \text{PGL}(3,q)$ of the Veronese surface. For each orbit, we provide an explicit representative $S$ and determine two combinatorial invariants: the point-orbit distribution and the hyperplane-orbit distribution. These invariants characterise the orbits except in two specific cases (in which the orbits are distinguished by their line-orbit distributions). In addition, we determine the stabiliser of $S$ in $K$, thereby obtaining the size of each orbit. As a consequence, we obtain a proof of the classification of pencils of conics in $\text{PG}(2,q)$, $q$ even, which to the best of our knowledge has been heretofore missing in the literature.

math.CO↗

arXiv2020

Combinatorial invariants for nets of conics in $\text{PG}(2,q)$

The problem of classifying linear systems of conics in projective planes dates back at least to Jordan, who classified pencils (one-dimensional systems) of conics over $\mathbb{C}$ and $\mathbb{R}$ in 1906--1907. The analogous problem for finite fields $\mathbb{F}_q$ with $q$ odd was solved by Dickson in 1908. In 1914, Wilson attempted to classify nets (two-dimensional systems) of conics over finite fields of odd characteristic, but his classification was incomplete and contained some inaccuracies. In a recent article, we completed Wilson's classification of nets of rank one, namely those containing a repeated line. The aim of the present paper is to introduce and calculate certain combinatorial invariants of these nets, which we expect will be of use in various applications. Our approach is geometric in the sense that we view a net of rank one as a plane in $\text{PG}(5,q)$ that meets the quadric Veronesean in at least one point; two such nets are then equivalent if and only if the corresponding planes belong to the same orbit under the induced action of $\text{PGL}(3,q)$ viewed as a subgroup of $\text{PGL}(6,q)$. We have previously determined the orbits of lines in $\text{PG}(5,q)$ under this action, which correspond to the aforementioned pencils of conics in $\text{PG}(2,q)$. The main contribution of this paper is to determine the line-orbit distribution of a plane $π$ corresponding to a net of rank one, namely, the number of lines in $π$ belonging to each line orbit. It turns out that this list of invariants completely determines the orbit of $π$, and we will use this fact in forthcoming work to develop an efficient algorithm for calculating the orbit of a given net of rank one. As a more immediate application, we also determine the stabilisers of nets of rank one in $\text{PGL}(3,q)$, and hence the orbit sizes.

math.CO↗

arXiv2020

Nets of conics of rank one in PG(2,q), q odd

We classify nets of conics in Desarguesian projective planes over finite fields of odd order, namely, two-dimensional linear systems of conics containing a repeated line. Our proof is geometric in the sense that we solve the equivalent problem of classifying the orbits of planes in $\text{PG}(5,q)$ which meet the quadric Veronesean in at least one point, under the action of $\text{PGL}(3,q) \leqslant \text{PGL}(6,q)$ (for $q$ odd). Our results complete a partial classification of nets of conics of rank one obtained by A. H. Wilson in the article "The canonical types of nets of modular conics", American Journal of Mathematics 36 (1914) 187-210.

math.CO↗

arXiv2019

The symmetric representation of lines in $\text{PG}(\mathbb{F}^3 \otimes \mathbb{F}^3)$

Let $\mathbb{F}$ be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space $V=\mathbb{F}^3 \otimes \mathbb{F}^3$ of $3 \times 3$ matrices over $\mathbb{F}$, and let $G \leq \text{PGL}(V)$ be the setwise stabiliser of the corresponding Segre variety $S_{3,3}(\mathbb{F})$ in the projective space $\text{PG}(V)$. The $G$-orbits of lines in $\text{PG}(V)$ were determined by the first author and Sheekey as part of their classification of tensors in $\mathbb{F}^2 \otimes V$ in the article "Canonical forms of $2 \times 3 \times 3$ tensors over the real field, algebraically closed fields, and finite fields", Linear Algebra Appl. 476 (2015) 133-147. Here we consider the related problem of classifying those line orbits that may be represented by {\em symmetric} matrices, or equivalently, of classifying the line orbits in the $\mathbb{F}$-span of the Veronese variety $\mathcal{V}_3(\mathbb{F}) \subset S_{3,3}(\mathbb{F})$ under the natural action of $K=\text{PGL}(3,\mathbb{F})$. Interestingly, several of the $G$-orbits that have symmetric representatives split under the action of $K$, and in many cases this splitting depends on the characteristic of $\mathbb{F}$. The corresponding orbit sizes and stabiliser subgroups of $K$ are also determined in the case where $\mathbb{F}$ is a finite field, and connections are drawn with old work of Jordan, Dickson and Campbell on the classification of pencils of conics in $\text{PG}(2,\mathbb{F})$, or equivalently, of pairs of ternary quadratic forms over $\mathbb{F}$.

math.CO↗

arXiv2017

On the second-largest Sylow subgroup of a finite simple group of Lie type

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm O}(\log_r(\ell)/\ell)}$, where $\ell$ is the Lie rank of $T$.

math.GR↗

arXiv2017

Simple groups, product actions, and generalised quadrangles

The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O'Nan-Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in non-Abelian finite simple groups, and about fixities of primitive permutation groups.

math.GR↗

arXiv2016

Point-primitive, line-transitive generalised quadrangles of holomorph type

Let $G$ be a group of collineations of a finite thick generalised quadrangle $Γ$. Suppose that $G$ acts primitively on the point set $\mathcal{P}$ of $Γ$, and transitively on the lines of $Γ$. We show that the primitive action of $G$ on $\mathcal{P}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples $Γ$ for which the action of $G$ on $\mathcal{P}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup $M$ and $M$ is non-Abelian.

math.GR↗

arXiv2015

Finding involutions with small support

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of $\varepsilon$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group.

math.GR↗