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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., Comment: 35 pages; minor formatting changes

The intersection graph $\Delta_G$ 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 Cs\'ak\'any and Poll\'ak in 1969. In this paper we answer two long-standing open questions posed by Cs\'ak\'any and Poll\'ak 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., Comment: 7 pages, comments welcome

The prime graph of a finite group $G$ is the labelled graph $\Gamma(G)$ 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 $\Gamma(H)=\Gamma(G)$ 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 $\Gamma(G)$, there are infinitely many groups $H$ with $\Gamma(H) \cong \Gamma(G)$., Comment: Various minor changes to the exposition; added Lemma 2.16; clarified proof of Lemma 4.2

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 $\mathbf{M}$., Comment: Merged with arXiv:2304.14646

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 classification of the maximal subgroups of $\mathbf{M}$. We construct two new maximal subgroups of $\mathbf{M}$, isomorphic to the automorphism groups of $\mathrm{PSL}_2(13)$ and $\mathrm{PSU}_3(4)$, and we show that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSL}_2(8)$ or $\mathrm{PSL}_2(16)$. In particular, we correct the claim that $\mathbf{M}$ has no almost simple maximal subgroup with socle $\mathrm{PSU}_3(4)$. 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, and we provide explicit generators for our newly discovered maximal subgroups of $\mathbf{M}$., Comment: This is intended to be the final version of the paper. It deals with all previously open and/or unpublished cases, i.e. PSU(3,4) and PSL(2,q) for q = 8, 13, and 16, and thereby completes the classification of the maximal subgroups of the Monster. Note that this version of the paper absorbs arXiv:2310.03317

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.

An irredundant base $B$ for a permutation group $G\leq \mathrm{Sym}(\Omega)$ is an ordered subset of $\Omega$ 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., Comment: 5 pages

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., Comment: 11 pages

Let $G$ be a finite permutation group on $\Omega$. An ordered sequence $(\omega_1,\ldots,\omega_\ell)$ of elements of $\Omega$ 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., Comment: 6 pages

Let $G$ be a permutation group on a set $\Omega$. A base for $G$ is a subset of $\Omega$ 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 $\Sigma(G)$ has vertex set $\Omega$ 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 $\Sigma(G)$ 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., Comment: Edited Remarks 1.3 and 1.4 for clarity; added link to Magma code; updated references

The prime graph, or Gruenberg--Kegel graph, of a finite group $G$ is the graph $\Gamma(G)$ 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 $\Gamma(H)=\Gamma(G)$ 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., Comment: Simplified the proofs for Co1 and M; fixed an error in the proof for B. Added more details of computations in all cases

Let $G$ be a finite non-regular primitive permutation group on a set $\Omega$ with point stabiliser $G_{\alpha}$. Then $G$ is said to be extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collection of groups. Moreover, if one assumes Wall's conjecture on the number of maximal subgroups of an almost simple group, then there is an explicit list of candidates, each of which has been eliminated in a recent paper by Burness and Thomas. So, modulo Wall's conjecture, the classification of extremely primitive groups is complete. In this paper we adopt a different approach, which allows us to complete this classification in full generality, independent of the veracity or otherwise of Wall's conjecture in the almost simple setting. Our method relies on recent work of Fawcett, Lee and others on the existence of regular orbits of almost simple groups acting on irreducible modules., Comment: 11 pages; to appear in the Israel Journal of Mathematics

Let $V$ be a finite-dimensional vector space over a finite field, and suppose $G \leq \Gamma \mathrm{L}(V)$ is a group with a unique subnormal quasisimple subgroup $E(G)$ that is absolutely irreducible on $V$. A base for $G$ is a set of vectors $B\subseteq V$ with pointwise stabiliser $G_B=1$. If $G$ has a base of size 1, we say that it has a regular orbit on $V$. In this paper we investigate the minimal base size of groups $G$ with $E(G)/Z(E(G)) \cong \mathrm{PSL}_n(q)$ in defining characteristic, with an aim of classifying those with a regular orbit on $V$., Comment: 45 pages

Let $G \leq \mathrm{GL}(V)$ be a group with a unique subnormal quasisimple subgroup $E(G)$ that acts absolutely irreducibly on $V$. A base for $G$ acting on $V$ is a set of vectors with trivial pointwise stabiliser in $G$. In this paper we determine the minimal base size of $G$ when $E(G)/Z(E(G))$ is a finite simple group of Lie type in cross-characteristic. We show that $G$ has a regular orbit on $V$, with specific exceptions, for which we find the base size., Comment: 54 pages. Fixed typos

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if $G$ is a quasisimple group (i.e. $G$ is perfect and $G/Z(G)$ is simple) acting irreducibly on $V$, then excluding two natural families, $G$ has a base of size at most 6. The two families consist of alternating groups ${\rm Alt}_m$ acting on the natural module of dimension $d = m-1$ or $m-2$, and classical groups with natural module of dimension $d$ over subfields of $F_q$.