A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfies $m(G) \leqslant a \cdot \delta (G)^b$, for $\delta (G) = \sum _{p \text { prime}} m(G_p)$, where $G_p$ is a Sylow p-subgroup of G. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$). In particular, we prove that there exist $a,b> 0$ such that any finite simple group G of Lie type of rank r over the field $\mathbb {F}_{p^f}$ satisfies $r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$, where $\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over $\mathbb {F}_{p^f}$ has size at most $ar^b + \omega (f)$.

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$. We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$, where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$. Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g.

Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$, we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$. The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $, for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$, which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$. We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a qth power in $\Gamma _{g}$, then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$

as $n\to \infty $, where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$.

Skew left braces arise naturally from the study of non-degenerate set-theoretic solutions of the Yang–Baxter equation. To understand the algebraic structure of skew left braces, a study of the decomposition into minimal substructures is relevant. We introduce chief series and prove a strengthened form of the Jordan–Hölder theorem for finite skew left braces. A characterization of right nilpotency and an application to multipermutation solutions are also given.

Improving and clarifying a construction of Horowitz and Shelah, we show how to construct (in $\mathsf {ZF}$, i.e., without using the Axiom of Choice) maximal cofinitary groups. Among the groups we construct, one is definable by a formula in second-order arithmetic with only a few natural number quantifiers.

Let G be a finite transitive group on a set $\Omega $, let $\alpha \in \Omega $, and let $G_{\alpha }$ be the stabilizer of the point $\alpha $ in G. In this paper, we are interested in the proportion

$$ \begin{align*} \frac{|\{\omega\in \Omega\mid \omega \textrm{ lies in a }G_{\alpha}\textrm{-orbit of cardinality at most 2}\}|}{|\Omega|}, \end{align*} $$

that is, the proportion of elements of $\Omega $ lying in a suborbit of cardinality at most 2. We show that, if this proportion is greater than $5/6$, then each element of $\Omega $ lies in a suborbit of cardinality at most 2, and hence G is classified by a result of Bergman and Lenstra. We also classify the permutation groups attaining the bound $5/6$.

We use these results to answer a question concerning the enumeration of Cayley graphs. Given a transitive group G containing a regular subgroup R, we determine an upper bound on the number of Cayley graphs on R containing G in their automorphism groups.

We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: ${\mathfrak{A}}_5$, ${\text{PSL}}_2(\textbf{F}_7)$, ${\mathfrak{A}}_6$, ${\text{SL}}_2(\textbf{F}_8)$, ${\mathfrak{A}}_7$, ${\text{PSp}}_4(\textbf{F}_3)$, ${\text{SL}}_2(\textbf{F}_{7})$, $2.{\mathfrak{A}}_5$, $2.{\mathfrak{A}}_6$, $3.{\mathfrak{A}}_6$ or $6.{\mathfrak{A}}_6$. All of these groups with a possible exception of $2.{\mathfrak{A}}_6$ and $6.{\mathfrak{A}}_6$ indeed act on some rationally connected threefolds.

We prove that any element in a sufficiently large transitive finite simple permutation group is a product of two derangements.

We determine all finite sets of equiangular lines spanning finite-dimensional complex unitary spaces for which the action on the lines of the set-stabiliser in the unitary group is 2-transitive with a regular normal subgroup.

We classify embeddings of the finite groups $A_4$, $S_4$ and $A_5$ in the Lie group $G_2(\mathbb C)$ up to conjugation.

Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

We give a sufficient and necessary condition for a Praeger–Xu graph to be a Cayley graph.

A group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.

If G is permutation group acting on a finite set $\Omega $, then this action induces a natural action of G on the power set $\mathscr{P}(\Omega )$. The number $s(G)$ of orbits in this action is an important parameter that has been used in bounding numbers of conjugacy classes in finite groups. In this context, $\inf ({\log _2 s(G)}/{\log _2 |G|})$ plays a role, but the precise value of this constant was unknown. We determine it where G runs over all permutation groups not containing any ${{\textrm {A}}}_l, l> 4$, as a composition factor.

A noncomplete graph is $2$-distance-transitive if, for $i \in \{1,2\}$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with the same distance i in the graph, there exists an element of the graph automorphism group that maps $(u_1,v_1)$ to $(u_2,v_2)$. This paper determines the family of $2$-distance-transitive Cayley graphs over dihedral groups, and it is shown that if the girth of such a graph is not $4$, then either it is a known $2$-arc-transitive graph or it is isomorphic to one of the following two graphs: $ {\mathrm {K}}_{x[y]}$, where $x\geq 3,y\geq 2$, and $G(2,p,({p-1})/{4})$, where p is a prime and $p \equiv 1 \ (\operatorname {mod}\, 8)$. Then, as an application of the above result, a complete classification is achieved of the family of $2$-geodesic-transitive Cayley graphs for dihedral groups.

Let $G$ be a primitive permutation group of degree $n$ with nonabelian socle, and let $k(G)$ be the number of conjugacy classes of $G$. We prove that either $k(G)< n/2$ and $k(G)=o(n)$ as $n\rightarrow \infty$, or $G$ belongs to explicit families of examples.

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

We show that, given a compact minimal system $(X,g)$ and an element h of the topological full group $\tau [g]$ of g, the infinite orbits of h admit a locally constant orientation with respect to the orbits of g. We use this to obtain a clopen partition of $(X,G)$ into minimal and periodic parts, where G is any virtually polycyclic subgroup of $\tau [g]$. We also use the orientation of orbits to give a refinement of the index map and to describe the role in $\tau [g]$ of the submonoid generated by the induced transformations of g. Finally, we consider the problem, given a homeomorphism h of the Cantor space X, of determining whether or not there exists a minimal homeomorphism g of X such that $h \in \tau [g]$.

A permutation group G on a set A is ${\kappa }$-homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$. G is ${\kappa }$-transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$.

Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ${\omega }$-homogeneous but not ${\omega }$-transitive permutation group on a cardinal ${\lambda }$ provided

1. (i) ${\lambda }<{\omega }_{\omega }$, or

2. (ii) $2^{\omega }<{\lambda }$, and ${\mu }^{\omega }={\mu }^+$ and $\Box _{\mu }$ hold for each ${\mu }\le {\lambda }$ with ${\omega }=\operatorname {cf}({\mu })<{{\mu }}$, or

3. (iii) our model was obtained by adding $(2^{\omega })^+$ many Cohen generic reals to some ground model.

For ${\kappa }>{\omega }$ we give a method to construct large ${\kappa }$-homogeneous, but not ${\kappa }$-transitive permutation groups. Using this method we show that there exist ${\kappa }^+$-homogeneous, but not ${\kappa }^+$-transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$.

A graph is edge-primitive if its automorphism group acts primitively on the edge set, and $2$-arc-transitive if its automorphism group acts transitively on the set of $2$-arcs. In this paper, we present a classification for those edge-primitive graphs that are $2$-arc-transitive and have soluble edge-stabilizers.