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A generalized quadrangle is a point-line incidence geometry Q such that: (i) any two points lie on at most one line, and (ii) given a line l and a point P not incident with l, there is a unique point of l collinear with P. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair (P, l ), and we say that the generalized quadrangle Q is antiflag- transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflag-transitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3, 5) or its dual. Our approach uses the theory of locally s-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291-317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187-233] on "large" subgroups of simple groups of Lie type to fully classify them. [ABSTRACT FROM AUTHOR]

We show that a complete bipartite graph Kpe, pf, where p is an odd prime, has an edge-transitive embedding in an orientable surface with all faces bounded by simple cycles if and only if e = f. There are exactly p2(e-1) such embeddings up to isomorphism. Among them, pe-1 are orientably regular, one of which is reflexible and pe-1 -1 form chiral pairs. The remaining p2(e-1) - pe-1 embeddings are non-regular (not arc-transitive). All of these embeddings have genus 1 2 (pe-1) (pe-2). [ABSTRACT FROM AUTHOR]

Let Γ be a self-complementary circulant of prime power order. It is shown that either F is a lexicographic product of two small self-complementary circulants or there exists a multiplicative automorphism of a regular cyclic subgroup that maps Γ to its complement. [ABSTRACT FROM AUTHOR]

Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l , where Σ = Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph. [ABSTRACT FROM AUTHOR]

Let Γ be a finite locally ( G, s)-arc transitive graph with s ≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a non-trivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the ‘basic’ graphs to study are those for which G acts quasiprimitively on at least one of the two orbits. In this paper we investigate the case where G is quasiprimitive on only one of the two G-orbits. Such graphs have a normal quotient which is a star. We construct several infinite families of locally 3-arc transitive graphs and prove characterization results for several of the possible quasiprimitive types for G. [ABSTRACT FROM AUTHOR]

Let $G$ and $X$ be transitive permutation groups on a set $\Omega$ such that $G$ is a normal subgroup of $X$. The overgroup $X$ induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of $G$ on $\Omega$. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where $X$ fixes no elements of $\operatorname{Orbl}(G,\Omega)$; such triples are called {\it exceptional}. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples $(G,X,\Omega,\mathcal{P})$ where $\mathcal{P}$ is a partition of $\operatorname{Orbl}(G,\Omega)$ such that $X$ is transitive on $\mathcal{P}$; such a quadruple is called a {\it TOD} (transitive orbital decomposition). It follows easily that the triple $(G,X,\Omega)$ in a TOD $(G,X,\Omega,\mathcal{P})$ is exceptional; conversely if an exceptional triple $(G,X,\Omega)$ is such that $X/G$ is cyclic of prime-power order, then there exists a partition $\mathcal{P}$ of $\operatorname{Orbl}(G,\Omega)$ such that $(G,X,\Omega,\mathcal{P})$ is a TOD. This paper characterizes TODs $(G,X,\Omega,\mathcal{P})$ such that $X^\Omega$ is primitive and $X/G$ is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs. [ABSTRACT FROM AUTHOR]

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms $G$ and are either locally $(G,s)$--arc transitive for $s \geq 2$ or $G$--locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of $G$. Given a normal subgroup $N$ which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of $N$ preserves both local primitivity and local $s$--arc transitivity and leads us to study graphs where $G$ acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for $G$ in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section. [ABSTRACT FROM AUTHOR]

A complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup. This solves a long-standing open problem in permutation group theory initiated by W. Burnside in 1900. [ABSTRACT FROM AUTHOR]

The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups are either solvable and act on a prime number of points, or are almost simple. Among the almost simple examples, the groups of Lie type have rank at most $2$, or the point stabilizer is a parabolic subgroup. [ABSTRACT FROM AUTHOR]

Let $G$ be a permutation group on a set $\Omega$ with a transitive normal subgroup $M$. Then $G$ acts on the set $\mathrm{Orbl}(M,\Omega)$ of nontrivial $M$-orbitals in the natural way, and here we are interested in the case where $\mathrm{Orbl}(M,\Omega)$ has a partition $\mathcal P$ such that $G$ acts transitively on $\mathcal P$. The problem of characterising such tuples $(M,G,\Omega,\mathcal P)$, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where $|\mathcal P|$ is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where $|\mathcal P|=2$ exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the $G$-actions on $\Omega$ and on $\mathcal P$, and gives some construction methods for TODs. [ABSTRACT FROM AUTHOR]

A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text{M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text{PSp}(4,p)$ with $p$ prime and $p\equiv\pm1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved. [ABSTRACT FROM AUTHOR]

Metallic dampers are widely used due to their satisfactory plastic energy dissipation, low price and convenient installation. In this study, an axial tension and compression corrugated steel plate damper (ATCCSPD) was proposed. According to the mechanical properties of the ATCCSPD, low-frequency cyclic loading tests and numerical verification of seven ATCCSPD specimens were designed and performed. Additionally, the test phenomena and failure modes of the ATCCSPD specimens were analysed, and seismic performance indices such as the hysteretic curve, skeleton curve, ductility and energy dissipation were obtained. Then, the impacts of the thickness of the corrugated web, the aspect ratio and the thickness of the flange on the seismic behaviour of the ATCCSPD specimens were investigated. The results demonstrated that the seven ATCCSPD specimens exhibited satisfactory ductility and energy dissipation capacity, and the hysteretic curves were stable and full. Overall, the ATCCSPD specimens exhibited axial tension and compression deformation, while the internal corrugated web exhibited pure shear deformation during the loading process. The web thickness and aspect ratio were the main factors affecting the bearing capacity and energy dissipation of the ATCCSPD specimens. The finite element software ABAQUS was used to simulate the ATCCSPD specimens, and shear damage was added to the nonlinear kinematic hardening constitutive model of steel, which could effectively simulate the failure process of the corresponding part after cracks formed in the ATCCSPD specimens. Finally, combined with the test results and the numerical results, the shear bearing capacity formula of the corrugated webs was fitted; then, the formula for calculating the tensile (compressive) bearing capacity of the ATCCSPD specimens was obtained by the superposition principle. By comparing the calculation results with the test results, it was found that the proposed formula was conservative and, hence, suitable for engineering applications. • An axial tension-compression corrugated steel plate damper (ATCCSPD) which can be replaced in the structure is proposed. • The impact of key parameters on the seismic performance of the ATCCSPDs were investigated. • The shear damage was added to the hardening model of steel could effectively simulate the failure process of the ATCCSPDs. • The new numerical simulation method could coincide with the test results. • The equation to determine the bearing capacity of the ATCCSPDs was proposed. [ABSTRACT FROM AUTHOR]