arXiv subjects

Melissa Lee

Publications and source records attributed to Melissa Lee.

At least 19 recordsLinked to original sources

arXiv2026

Finding 59:29 in the Monster

The classification of the maximal subgroups of the Monster group has been completed recently, and explicit generators for each such subgroup (up to conjugacy) have been made available for the software mmgroup, with the exception of the maximal subgroup 59:29, see Dietrich et al. (Adv. Math., 2025; J Algebra, 2026). We provide explicit generators for this last maximal subgroup and comment on the extensive search that led to finding them. Our method is similar to, but significantly more involved than Bray et al.'s (London Math. Soc. J. Comput. Math., 2016) approach for constructing 47:23 in the Baby Monster. Our result allows us to provide a new short proof that the Monster does not have a subgroup PSL2(59), correcting a result of Holmes and Wilson (J. London Math. Soc., 2004).

math.GR↗

arXiv2026

An infinite family of counterexamples to the Polycirculant Conjecture

We disprove the Polycirculant Conjecture, which states that every transitive 2-closed permutation group is non-elusive, i.e. contains a derangement of prime order. In fact, we prove a stronger result, answering a long-standing question of Maru\v{s}i\v{c} and Jordan: there exists a vertex-transitive graph admitting no semiregular automorphism. To do so, we employ recently developed methods of Chen et al. for constructing elusive groups via non-split extensions, allowing us to construct an elusive group $7^6.\mathrm{PSU}_3(3)$ of degree 16,464. We show that this group is the full automorphism group of seven of its orbital graphs and hence is 2-closed. Our example extends to infinitely many counterexamples of the Polycirculant Conjecture, and infinitely many vertex-transitive graphs admitting no semiregular automorphism.

math.GR↗

arXiv2026

On the trivial units property and the unique product property

We report on some computational experiments related to the trivial units property and unique product property for group rings of torsion-free groups. These properties are related to Kaplansky's unit and zero-divisor conjectures. Our investigations include a classification of certain symmetric non-trivial units in the binary group ring of the Hantzsche-Wendt group; this group was used in Gardam's refutal of Kaplansky's unit conjecture. We also exhibit and investigate a new candidate group that fails the unique units property but may satisfy the trivial unit property. No examples of groups with these properties are known to date.

math.GR↗

arXiv2026

Automorphisms of Kimura Hadamard Matrices

We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms, and show that the automorphism group of a KHM always has a subgroup isomorphic to $D_{2k}\times Q_8$, or $C_2\times D_{2k}\times Q_8$ if it is $y$-invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda--Yamada construction, reveals counterexamples to a conjecture of \'O Catha\'{\i}n and suggests that no further automorphisms occur beyond those predicted by our framework.

math.CO↗

arXiv2025

Elusive groups from non-split extensions

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is $2$-closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. As a demonstration, we construct elusive groups of new degrees, namely $p^{3k-4}(p+1)/2$ for each Mersenne prime $p\geq7$ and integer $k\geq2$. We also construct the first examples of elusive groups with odd degree, namely $3^{k+1}\cdot5^2$, and twice odd degree, namely $2\cdot3^{k + 1}\cdot5^2$ for each $k\geq1$. We conclude by proposing further problems to advance this new direction of research.

math.GR↗

arXiv2025

Prime simplicial complexes of finite groups

The prime simplicial complex $Π(G)$ of a finite group $G$ is composed of all sets of primes $S$ where $G$ has an element of order the product of primes in $S$, with the subsets partially ordered by inclusion. This complex was introduced by Peter Cameron as the generalisation of the well-studied prime (or Gruenberg-Kegel) graphs. In this paper, we establish new results concerning two key properties of $Π(G)$: recognisability and purity. We demonstrate that recognisability by the prime simplicial complex is strictly stronger than recognisability by the prime graph. Notably, we present the first known example of a group that is recognisable by its prime simplicial complex and spectrum, but not by its prime graph, and is not a direct product of two isomorphic simple groups. Furthermore, we classify groups with pure prime simplicial complexes (i.e., all maximal simplices have the same size) across several infinite families of finite simple groups. We also provide a partial classification for non-abelian finite simple groups whose prime simplicial complex has maximal simplices of size at most 2.

math.GR↗

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↗

arXiv2025

The Saxl hypergraph of a permutation group

Given a permutation group $G \le \mathrm{Sym}(Ω)$, a subset $B$ of $Ω$ is said to be a base if its pointwise stabiliser in $G$ is trivial, and the base size $b(G)$ is the minimum size of a base. In the notable case $b(G) = 2$, Burness and Giudici define the Saxl graph of $G$ to be the graph on $Ω$ with bases of size 2 as edges. Later work of Freedman et al. extends this notion to any group for which $b(G) \ge 2$, taking the pairs of points contained in bases of size $b(G)$ for edges. We study an alternative generalisation, the Saxl hypergraph, where bases of size $b(G)$ are themselves the edges. In particular, we consider groups with complete Saxl hypergraphs, primitive groups whose Saxl hypergraphs have flag-spanning tours, and appropriate generalisations of Burness and Giudici's Common Neighbour Conjecture.

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

On the generalised Saxl graphs of permutation groups

A base for a finite permutation group $G \le \mathrm{Sym}(\Omega)$ is a subset of $\Omega$ with trivial pointwise stabiliser in $G$, and the base size of $G$ is the smallest size of a base for $G$. Motivated by the interest in groups of base size two, Burness and Giudici introduced the notion of the Saxl graph. This graph has vertex set $\Omega$, with edges between elements if they form a base for $G$. We define a generalisation of this graph that encodes useful information about $G$ whenever $b(G) \ge 2$: here, the edges are the pairs of elements of $\Omega$ that can be extended to bases of size $b(G)$. In particular, for primitive groups, we investigate the completeness and arc-transitivity of the generalised graph, and the generalisation of Burness and Giudici's Common Neighbour Conjecture on the original Saxl graph.

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

On the diameter of intersection graphs of finite groups

The intersection graph $Δ_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Csákány and Pollák in 1969. In this paper we answer two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.

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↗

arXiv2023

Using online student focus groups in the development of new educational resources

Educational resources, such as web apps and self-directed tutorials, have become popular tools for teaching and active learning. Ideally, students - the intended users of these resources - should be involved in the resource development stage. However, in practice students often only interact with fully developed resources, when it might be too late to incorporate changes. Previous work has addressed this by involving students in the development of new resources via in-person focus groups and interviews. In these, the resource developers observe students interacting with the resource. This allows developers to incorporate their observations and students' direct feedback into further development of the resource. However, as a result of the COVID-19 pandemic, carrying out in-person focus groups became infeasible due to social distancing restrictions. Instead, online meetings and classes became ubiquitous. In this work, we describe a fully-online methodology to evaluate new resources in development. Specifically, our methodology consists of carrying out student focus groups via online video conferencing software. We assessed two educational resources for introductory statistics using our methodology and found that the online setting allowed us to obtain rich, detailed information from the students. We also found online focus groups to be more efficient: students and researchers did not need to travel and scheduling was not restricted by the availability of physical space. Our findings suggest that online focus groups are an attractive alternative to in-person focus groups for student assessment of resources in development, even now that pandemic restrictions are being eased.

stat.OT↗

arXiv2023

Primitive almost simple IBIS groups with sporadic socle

An irredundant base $B$ for a permutation group $G\leq \mathrm{Sym}(Ω)$ is an ordered subset of $Ω$ with trivial stabiliser such that no base point is fixed by the stabiliser of its predecessors. Groups whose irredundant bases all have the same size are termed Irredundant Bases of Invariant Size (IBIS) groups, and were introduced by Cameron and Fon-Der-Flaass. In this paper, we contribute to the classification of primitive IBIS groups by classifying those that are almost simple with sporadic socle.

math.GR↗

arXiv2022

Extremely primitive groups and linear spaces

A finite non-regular primitive permutation group $G$ is extremely primitive if a point stabiliser acts primitively on each of its nontrivial orbits. Such groups have been studied for almost a century, finding various applications. The classification of extremely primitive groups was recently completed by Burness and Lee, who relied on an earlier classification of soluble extremely primitive groups by Mann, Praeger and Seress. Unfortunately, there is an inaccuracy in the latter classification. We correct this mistake, and also investigate regular linear spaces which admit groups of automorphisms that are extremely primitive on points.

math.CO↗

arXiv2022

A classification of finite primitive IBIS groups with alternating socle

Let $G$ be a finite permutation group on $Ω$. An ordered sequence $(ω_1,\ldots,ω_\ell)$ of elements of $Ω$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups $G$ to the case that the socle of $G$ is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

math.GR↗