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Cai Heng Li

Publications and source records attributed to Cai Heng Li.

At least 55 records · Page 3Linked to original sources
arXiv2020

A classification of finite locally 2-transitive generalized quadrangles

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $π$ acts $2$-transitively on the points of $π$, then $π$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{PΓL}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.

math.GR↗
arXiv2019

Skew-morphisms of nonabelian characteristically simple groups

A skew-morphism of a finite group $G$ is a permutation $\s$ on $G$ fixing the identity element, and for which there exists an integer function $π$ on $G$ such that $\s(xy)=\s(x)\s^{π(x)}(y)$ for all $x,y\in G$. It has been known that given a skew-morphism $\s $ of $G$, the product of $\lg \s \rg$ with the left regular representation of $G$ forms a permutation group on $G$, called the skew-product group of $\s$. The skew-morphism was introduced as an algebraic tool to investigate regular Cayley maps. In this paper, the skew-product groups are characterized, for all skew-morphisms of finite nonabelian characteristically simple groups (see Theorem 1.1) and correspondingly the Cayley maps on these groups are characterized (see Theorem 1.5).

math.CO↗
arXiv2019

Constructing 2-Arc-Transitive Covers of Hypercubes

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.

math.CO↗
arXiv2019

Vertex-primitive $s$-arc-transitive digraphs of linear groups

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.

math.CO↗
arXiv2017

The number of composition factors of order $p$ in completely reducible groups of characteristic $p$

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.

math.GR↗
arXiv2016

Arc-transitive digraphs with quasiprimitive local actions

Let $Γ$ be a finite $G$-vertex-transitive digraph. The in-local action of $(Γ,G)$ 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.

math.GR↗
arXiv2016

Cubic arc-transitive $k$-circulants

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 $6$.

math.CO↗
arXiv2016

Factorizations of almost simple groups with a solvable factor, and Cayley graphs of solvable groups

A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the cycles, every non-bipartite connected 3-arc-transitive Cayley graph of a solvable group is a cover of the Petersen graph or the Hoffman-Singleton graph.

math.GR↗
arXiv2015

A classification of finite antiflag-transitive generalized quadrangles

A generalized quadrangle is a point-line incidence geometry $\mathcal{Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz (1973), and we study a larger class of generalized quadrangles: the \emph{antiflag-transitive} quadrangles. An antiflag of a generalized quadrangle is a non-incident point-line pair $(P, \ell)$, and we say that the generalized quadrangle $\mathcal{Q}$ 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 $\mathcal{Q}$ is antiflag-transitive, then $\mathcal{Q}$ is either a classical generalized quadrangle or is the unique generalized quadrangle of order $(3,5)$ or its dual.

math.CO↗
arXiv2015

Finite 2-geodesic transitive graphs of prime valency

We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $Γ$ is 2-arc transitive or the valency $p$ satisfies $p\equiv 1\pmod 4$, and for each such prime there is a unique graph with this property: it is a non-bipartite antipodal double cover of the complete graph $K_{p+1}$ with automorphism group $PSL(2,p)\times Z_2$ and diameter 3.

math.CO↗
arXiv2013

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give a general construction for this case.

math.CO↗
arXiv2013

Characterising star-transitive and st(edge)-transitive graphs

Recent work of Lazarovich provides necessary and sufficient conditions on a graph L for there to exist a unique simply-connected (k,L)-complex. The two conditions are symmetry properties of the graph, namely star-transitivity and st(edge)-transitivity. In this paper we investigate star-transitive and st(edge)-transitive graphs by studying the structure of the vertex and edge stabilisers of such graphs. We also provide new examples of graphs that are both star-transitive and st(edge)-transitive.

math.CO↗
arXiv2012

Line graphs and $2$-geodesic transitivity

For a graph $Γ$, a positive integer $s$ and a subgroup $G\leq \Aut(Γ)$, we prove that $G$ is transitive on the set of $s$-arcs of $Γ$ if and only if $Γ$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive.

math.CO↗
arXiv2011

On distance, geodesic and arc transitivity of graphs

We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of $s$, and that these graphs can have arbitrarily large diameter if and only if $1\leq s\leq 3$. Moreover, for a prime $p$ we prove that there exists a graph of valency $p$ that is 2-geodesic transitive but not 2-arc transitive if and only if $p\equiv 1\pmod 4$, and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph $K_{p+1}$ and is geodesic transitive with automorphism group $PSL(2,p)\times Z_2$.

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