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A digraph is $s$-arc-transitive if its automorphism group is transitive on directed paths with $s$ edges, that is, on $s$-arcs. Although infinite families of finite $s$-arc transitive digraphs of arbitrary valency were constructed by the third author in 1989, existence of a vertex-primitive $2$-arc-transitive digraph was not known until an infinite family was constructed by the second author with Li and Xia in 2017. This led to a conjecture by the second author and Xia in 2018 that, for a finite vertex-primitive $s$-arc-transitive digraph, $s$ is at most $2$, together with their proof that it is sufficient to prove the conjecture for digraphs with an almost simple group of automorphisms. This paper confirms the conjecture for finite symplectic groups.

Suppose we have a finite thick generalised quadrangle whose automorphism group $G$ acts primitively on both the set of points and the set of lines. Then $G$ must be almost simple. In this paper, we show that $\operatorname{soc}(G)$ cannot be isomorphic to $\operatorname{Sz}(2^{2m+1})$ or $\operatorname{Ree}(3^{2m+1})$ where $m$ is a positive integer., Comment: 11 pages

For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of \emph{synchronising} and \emph{separating} are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of \emph{spreading} permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an \emph{AB-Lemma}) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

In this paper we extend the construction of Giudici, Morgan and Zhou [arXiv:2110.07896] to give the first known examples of nonregular, $2$-closed permutation groups of rank greater than $4$ that are not the automorphism group of any digraph. We also show that this construction only gives examples for four particular primes.

The synchronisation hierarchy of finite permutation groups consists of classes of groups lying between 2-transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabiliser is supplemented by all corresponding two-point stabilisers., Comment: 10 pages. Version 2 resolves the Monster group case of Theorem 1.3, with the aid of a result drawn to our attention by Prof. Tim Burness. Version 3 corrects an entry in the |T:A| column of Table 1. To appear in Proc. Roy. Soc. Edinburgh Sect. A

We classify, up to some notoriously hard cases, the rank 3 graphs which fail to meet either the Delsarte or the Hoffman bound. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give new examples of synchronising, but not $\mathbb{Q}\mathrm{I}$, groups of affine type., Comment: Added acknowledgements and noted the third author's change of institution

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf N}_G (\langle x\rangle){\bf N}_G(\langle y\rangle)$, where we are denoting by ${\bf N}_G(\langle x\rangle)$ and by ${\bf N}_G(\langle y\rangle)$ the normalizers of the cyclic subgroups $\langle x\rangle$ and $\langle y\rangle$. As a consequence of this result, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf C}_G(x){\bf C}_G(y)$., Comment: 36 pages

For a positive integer $k$, a group $G$ is said to be totally $k$-closed if for each set $\Omega$ upon which $G$ acts faithfully, $G$ is the largest subgroup of $\mathrm{Sym}(\Omega)$ that leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\cdots\times \Omega=\Omega^k$. Each finite group $G$ is totally $|G|$-closed, and $k(G)$ denotes the least integer $k$ such that $G$ is totally $k$-closed. We address the question of determining the closure number $k(G)$ for finite simple groups $G$. Prior to our work it was known that $k(G)=2$ for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $k(G)\geq3$ for all other finite simple groups. We determine the value for the alternating groups, namely $k(A_n)=n-1$. In addition, for all simple groups $G$, other than alternating groups and classical groups, we show that $k(G)\leq 7$. Finally, if $G$ is a finite simple classical group with natural module of dimension $n$, we show that $k(G)\leq n+2$ if $n \ge 14$, and $k(G) \le \lfloor n/3 + 12 \rfloor$ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on $n$ and the type of $G$) on the base sizes of the primitive actions of $G$, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups., Comment: 18 pages. Minor edits, plus a finiteness condition added to Lemma 5.1

We study locally $s$-arc-transitive graphs arising from the quasiprimitive product action (PA). We prove that, for any locally $(G,2)$-arc-transitive graph with $G$ acting quasiprimitively with type PA on both $G$-orbits of vertices, the group $G$ does not act primitively on either orbit. Moreover, we construct the first examples of locally $s$-arc-transitive graphs of PA type that are not standard double covers of $s$-arc-transitive graphs of PA type, answering the existence question for these graphs.

Determining an upper bound on $s$ for finite vertex-primitive $s$-arc-transitive digraphs has received considerable attention dating back to a question of Praeger in 1990. It was shown by Giudici and Xia that the smallest upper bound on $s$ is attained for some digraph admitting an almost simple $s$-arc-transitive group. In this paper, based on the work of Pan, Wu and Yin, we prove that $s\leqslant 2$ in the case where the group is an alternating or symmetric group.

We characterise the primitive 2-closed groups $G$ of rank at most four that are not the automorphism group of a graph or digraph and show that if the degree is at least 2402 then there are just two infinite families or $G\leqslant \mathrm{A}\Gamma\mathrm{L}_1(p^d)$, the 1-dimensional affine semilinear group. These are the first known examples of non-regular 2-closed groups that are not the automorphism group of a graph or digraph.

The investigation of s-arc-transitivity of digraphs can be dated back to 1989 when the third author showed that s can be arbitrarily large if the action on vertices is imprimitive. However, the situation is completely different when the digraph is vertex-primitive and not a directed cycle. In 2017 the second author, Li and Xia constructed the first infinite family of G-vertex-primitive 2-arc-transitive examples, and asked if there is an upper bound on s for G-vertex-primitive s-arc-transitive digraphs w=that are not directed. In 2018 the second author and Xia showed that if there is a largest such value of s then it will occur when G is almost simple. So far it has been shown that s\leq 2 for almost simple groups whose socle is an alternating group or a projective special linear group. The contribution of this paper is to prove that s\leq 1 in the case of the Suzuki and the small Ree groups. We give constructions with s=1 to show that the bound is sharp.

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

We say that a finite group $G$ acting on a set $\Omega$ has Property $(*)_p$ for a prime $p$ if $P_\omega$ is a Sylow $p$-subgroup of $G_\omega$ for all $\omega\in\Omega$ and Sylow $p$-subgroups $P$ of $G$. Property $(*)_p$ arose in the recent work of Tornier (2018) on local Sylow $p$-subgroups of Burger-Mozes groups, and he determined the values of $p$ for which the alternating group $A_n$ and symmetric group $S_n$ acting on $n$ points has Property $(*)_p$. In this paper, we extend this result to finite $2$-transitive groups and we give a structural characterisation result for the finite primitive groups that satisfy Property $(*)_p$ for an allowable prime $p$., Comment: minor corrections

For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles., Comment: 62 pages, 34 tables

We study finite transitive permutation groups $G\leqslant\operatorname{Sym}(\Omega)$ such that all orbits of the conjugation action on $G$ of the normaliser of $G$ in $\operatorname{Sym}(\Omega)$ have size bounded by some constant. Our results extend recent results, due to the first author, on finite abstract groups $G$ such that all orbits of the natural action of the automorphism group $\operatorname{Aut}(G)$ on $G$ have size bounded by some constant., Comment: 23 pages; reupload after PDF creation from v1 failed

We examine the subgroup $D(G)$ of a transitive permutation group $G$ which is generated by the derangements in $G$. Our main results bound the index of this subgroup: we conjecture that, if $G$ has degree $n$ and is not a Frobenius group, then $|G:D(G)|\leqslant\sqrt{n}-1$; we prove this except when $G$ is a primitive affine group. For affine groups, we translate our conjecture into an equivalent form regarding $|H:R(H)|$, where $H$ is a linear group on a finite vector space and $R(H)$ is the subgroup of $H$ generated by elements having eigenvalue~$1$. If $G$ is a Frobenius group, then $D(G)$ is the Frobenius kernel, and so $G/D(G)$ is isomorphic to a Frobenius complement. We give some examples where $D(G)\ne G$, and examine the group-theoretic structure of $G/D(G)$; in particular, we construct groups $G$ in which $G/D(G)$ is not a Frobenius complement.

We give a comprehensive description of the functions and variables defined in the authors' GAP code file OrbOrd.txt, which serve mainly to compute (bounds on) the number of $\operatorname{Aut}(S)$-orbits on $S$, or the set or number of element orders in $S$ for nonabelian finite simple groups of Lie type $S$., Comment: 46 pages

For a finite group $G$, we denote by $\omega(G)$ the number of $\operatorname{Aut}(G)$-orbits on $G$, and by $\operatorname{o}(G)$ the number of distinct element orders in $G$. In this paper, we are primarily concerned with the two quantities $\mathfrak{d}(G):=\omega(G)-\operatorname{o}(G)$ and $\mathfrak{q}(G):=\omega(G)/\operatorname{o}(G)$, each of which may be viewed as a measure for how far $G$ is from being an AT-group in the sense of Zhang (that is, a group with $\omega(G)=\operatorname{o}(G)$). We show that the index $|G:\operatorname{Rad}(G)|$ of the soluble radical $\operatorname{Rad}(G)$ of $G$ can be bounded from above both by a function in $\mathfrak{d}(G)$ and by a function in $\mathfrak{q}(G)$ and $\operatorname{o}(\operatorname{Rad}(G))$. We also obtain a curious quantitative characterisation of the Fischer-Griess Monster group $\operatorname{M}$., Comment: 98 pages; incorporated referee comments in v2; to appear in Mem. Amer. Math. Soc

This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.

A Cayley graph is said to be an NNN-graph if it is both normal and non-normal for isomorphic regular groups, and a group has the NNN-property if there exists an NNN-graph for it. In this paper we investigate the NNN-property of cyclic groups, and show that cyclic groups do not have the NNN-property.

In this paper, we characterise the family of finite arc-transitive bicirculants. We show that every finite arc-transitive bicirculant is a normal $r$-cover of an arc-transitive graph that lies in one of eight infinite families or is one of seven sporadic arc-transitive graphs. Moreover, each of these "basic" graphs is either an arc-transitive bicirculant or an arc-transitive circulant, and each graph in the latter case has an arc-transitive bicirculant normal $r$-cover for some integer $r$.

We prove that every finite arc-transitive graph of valency twice a prime admits a nontrivial semiregular automorphism, that is, a non-identity automorphism whose cycles all have the same length. This is a special case of the Polycirculant Conjecture, which states that all finite vertex-transitive digraphs admit such automorphisms.

Distance-regular graphs have many beautiful combinatorial properties. Distance-transitive graphs have very strong symmetries, and they are distance-regular, i.e. distance-transitivity implies distance-regularity. In this paper, we give similar results, i.e. for special $s$ and graphs with other restrictions we show that $s$-distance-transitivity implies distance-regularity., Comment: 34 pages

We introduce the notion of a symmetric basis of a vector space equipped with a quadratic form, and provide a sufficient and necessary condition for the existence to such a basis. Symmetric bases are then used to study Cayley graphs of certain extraspecial 2-groups of order 2^{2r+1} (r\geq 1), which are further shown to be normal Cayley graphs and 2-arc-transitive covers of 2r-dimensional hypercubes.

Dolfi, Guralnick, Praeger and Spiga asked if there exist infinitely many primitive groups of twisted wreath type with nontrivial coprime subdegrees. Here we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with nontrivial coprime subdegrees. In particular, we define a primitive twisted wreath group $G(m,q)$ constructed from the nonabelian simple group $\text{PSL}(2,q)$ and a primitive permutation group of diagonal type with socle $\text{PSL}(2,q)^m$, and determine all values of $m$ and $q$ for which $G(m,q)$ has nontrivial coprime subdegrees. In the case where $m=2$ and $q\notin\{7,11,29\}$ we obtain a full classification of all pairs of nontrivial coprime subdegrees.

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabiliser in $G$ is trivial. In this paper we introduce and study an associated graph $\Sigma(G)$, which we call the Saxl graph of $G$. The vertices of $\Sigma(G)$ are the points of $\Omega$, and two vertices are adjacent if they form a base for $G$. This graph encodes some interesting properties of the permutation group. We investigate the connectivity of $\Sigma(G)$ for a finite transitive group $G$, as well as its diameter, Hamiltonicity, clique and independence numbers, and we present several open problems. For instance, we conjecture that if $G$ is a primitive group with a base of size $2$, then the diameter of $\Sigma(G)$ is at most $2$. Using a probabilistic approach, we establish the conjecture for some families of almost simple groups. For example, the conjecture holds when $G=S_n$ or $A_n$ (with $n>12$) and the point stabiliser of $G$ is a primitive subgroup. In contrast, we can construct imprimitive groups whose Saxl graph is disconnected with arbitrarily many connected components, or connected with arbitrarily large diameter., Comment: 27 pages; to appear in Math. Proc. Cambridge Philos. Soc

We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.

We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.

Let $\Gamma$ be a finite $G$-vertex-transitive digraph. The in-local action of $(\Gamma,G)$ is the permutation group $L_-$ induced by the vertex-stabiliser on the set of in-neighbours of $v$. The out-local action $L_+$ is defined analogously. Note that $L_-$ and $L_+$ may not be isomorphic. We thus consider the problem of determining which pairs $(L_-,L_+)$ are possible. We prove some general results, but pay special attention to the case when $L_-$ and $L_+$ are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other.

Let $G$ be a transitive permutation group of degree $n$ with point stabiliser $H$ and let $r$ be a prime divisor of $n$. We say that $G$ is $r$-elusive if it does not contain a derangement of order $r$. The problem of determining the $r$-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of $r$-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining non-geometric actions where $H$ is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups., Comment: 39 pages; to appear in Israel J. Math

We solve the long-standing existence problem of vertex-primitive 2-arc-transitive digraphs by constructing an infinite family of such digraphs.

A transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular. Interest in this class of groups is motivated by two sources: problems arising in universal algebra related to collapsing monoids and the graph-restrictive problem for permutation groups. Here we develop a theory of semiprimitive groups which encompasses their structure, their quotient actions and a method by which all finite semiprimitive groups are constructed. We also extend some results from the theory of primitive groups to semiprimitive groups, and conclude with open problems of a similar nature., Comment: 34 pages

We classify finite primitive permutation groups having a suborbit of length 5. As a corollary, we obtain a classification of finite vertex-primitive graphs of valency 5. In the process, we also classify finite almost simple groups that have a maximal subgroup isomorphic to $\mathrm{Alt}(5)$ or $\mathrm{Sym}(5)$.

For an integer $k\geq 1$, a graph is called a $k$-circulant if its automorphism group contains a cyclic semiregular subgroup with $k$ orbits on the vertices. We show that, if $k$ is even, there exist infinitely many cubic arc-transitive $k$-circulants. We conjecture that, if $k$ is odd, then a cubic arc-transitive $k$-circulant has order at most $6k^2$. Our main result is a proof of this conjecture when $k$ is squarefree and coprime to $6$.

Let $q$ be a power of a prime $p$ and let $G$ be a completely reducible subgroup of $\mathrm{GL}(d,q)$. We prove that the number of composition factors of $G$ that have prime order $p$ is at most $(\varepsilon_q d-1)/(p-1)$, where $\varepsilon_q$ is a function of $q$ satisfying $1\leqslant\varepsilon_q\leqslant 3/2$. For every $q$, we give examples showing this bound is sharp infinitely often., Comment: 14 pages; Theorem 1 strengthened if $G$ is reducible; Corollary 10 added; reference [6] added

Let $G$ be a transitive permutation group of degree $n$. We say that $G$ is $2'$-elusive if $n$ is divisible by an odd prime, but $G$ does not contain a derangement of odd prime order. In this paper we study the structure of quasiprimitive and biquasiprimitive $2'$-elusive permutation groups, extending earlier work of Giudici and Xu on elusive groups. As an application, we use our results to investigate automorphisms of finite arc-transitive graphs of prime valency., Comment: 21 pages; to appear in J. Combin. Theory Ser. A

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs $\Gamma$ that admit such a group $G$ as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which $G_0$ is a subgroup of either ${\rm \Gamma L}(n,q)$ or ${\rm \Gamma Sp}(n,q)$ and is maximal in one of the Aschbacher classes $\mathcal{C}_i$, where $i \in \{2,4,5,6,7,8\}$. We are able to determine all graphs $\Gamma$ which arise from $G_0 \leq {\rm \Gamma L}(n,q)$ with $i \in \{2,4,8\}$, and from $G_0 \leq {\rm \Gamma Sp}(n,q)$ with $i \in \{2,8\}$. For the remaining classes we give necessary conditions in order for $\Gamma$ to have diameter two, and in some special subcases determine all $G$-symmetric diameter two graphs.

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain a complete classification of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.