arXiv subjects
Publications and source records attributed to Cai Heng Li.
A classification is given of regular, rotary, and birotary Cayley maps of which the vertex number is coprime to the valency. The classification is involved in constructions of new examples of interesting Cayley maps.
The classification of factorizations $G=HK$ of finite almost simple groups, proposed by Wielandt in 1979 and pursued through several partial classifications, has remained open in one major case. We settle that case: for every finite almost simple classical group $G$, we determine all factorizations $G=HK$ in which both $H$ and $K$ have a unique nonsolvable composition factor, completing the classification of factorizations of finite almost simple groups. Among other consequences of this classification, we prove that the smallest dimension of a biperfect bicrossproduct Hopf algebra is $41287680$.
This paper presents a classification of locally $2$-homogeneous designs, extending Kantor's classification of 2-transitive symmetric designs (1985).
We construct connected $2$-arc-transitive covers of complete graphs with non-abelian characteristically simple transformation groups. This solves the existence problem for non-solvable $2$-arc-transitive covers of complete graphs.
A classification is given of finite $k$-set-homogeneous graphs for $k\geqslant 2$, leading to a striking result that each finite $k$-set-homogeneous graph is $k$-homogeneous. It shows that $3$-set-homogeneous graphs are rare, consisting of the following graphs and their complements: $\C_5$, $\K_n\square\K_n$, $n\K_m$, the Schläfli graph of order 27, the Higman-Sims graph, the MaLaughlin graph, {affine polar graphs, and elliptic orthogonal graphs}. As an ingredient for the proof, it is shown that all orbitals in a primitive permutation group of rank $4$ are self-paired, except for $\PSU_3(3)$ acting on 36 points.
A characterization is given of finite groups $H$ that have skew-morphisms of order coprime to the order $|H|$, and their skew-morphisms. A complete classification is then given of the automorphism groups and the underlying graphs of vertex-rotary core-free Hall Cayley maps.
This is one of a series of papers which aims towards a classification of imprimitive affine groups of rank $3$. In this paper, a complete classification is given of such groups of characteristic $p$ such that the point stabilizer is not $p$-local, which shows that such groups are very rare, namely, the two non-isomorphic groups of the form $2^4{:}\mathrm{GL}_3(2)$ with a unique minimal normal subgroup are the only examples.
A map is bi-orientable if it admits an assignment of local orientations to its vertices such that for every edge, the local orientations at its two endpoints are opposite. Such an assignment is called a bi-orientation of the map. A bi-orientable map is bi-rotary if its automorphism group contains an arc-regular subgroup that preserves the bi-orientation. In this paper, we characterize the automorphism group structure of bi-rotary maps whose Euler characteristic is a negative prime power.
We give an explicit characterization of solvable factors in factorizations of finite classical groups of Lie type. This completes the classification of solvable factors in factorizations of almost simple groups, finishing the program initiated in [Memoirs of the AMS, 279 (2022), no.~1375] and [Advances in Mathematics, 377 (2021), 107499]. In particular, it resolves the final remaining case in the long-standing problem of determining exact factorizations of almost simple groups. As a byproduct, we obtain a new characterization of one-dimensional transitive groups, offering further insights into their group structures. We also apply our classification to describe quasiprimitive permutation groups with a solvable transitive subgroup, leading to an interesting result that these subgroups are ``small''.
We construct connected $2$-arc-transitive covers of the Petersen graph with non-solvable transformation groups, solving the long-standing problem for the existence of such covers.
A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately transitive groups.The latter three classes of groups of rank $3$ have been classified, forming significant progresses on the long-standing problem of classifying permutation groups of rank $3$.In this paper, a complete classification is given of finite semiprimitive groups of rank $3$ that are not innately transitive, examples of which are certain Schur coverings of certain almost simple $2$-transitive groups, and three exceptional small groups.
Let $G\leqslant\mathrm{Sym}(Ω)$ be a finite transitive permutation group with point stabiliser $H$. We say that a subgroup $K$ of $G$ is a fixer if every element of $K$ has fixed points, and we say that $K$ is large if $|K| \geqslant |H|$. There is a special interest in studying large fixers due to connections with Erdős-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle $\mathrm{PSL}_2(q)$, and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group.
A complete classification is given of finite groups whose elements are partitioned into three orbits by the automorphism groups, solving the long-standing classification problem initiated by G. Higman in 1963. As a consequence, a classification is obtained for finite permutation groups of rank $3$ which are holomorphs of groups.
This is the second of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one carries out the classification work for arc-transitive maps with non-solvable automorphism groups, which together with the first one completes a description of arc-transitive maps with the Euler characteristic and the edge number coprime. The classification is involved with a construction of some new and interesting reversing maps.
This is one of a series of papers which aim towards a classification of edge-transitive maps of which the Euler characteristic and the edge number are coprime. This one establishes a framework and carries out the classification work for arc-transitive maps with solvable automorphism groups, which illustrates how the edge number impacts on the Euler characteristic for maps. The classification is involved with the constructions of various new and interesting arc-regular maps.
This paper presents a classification of exact factorizations of almost simple groups, which has been a long-standing open problem initiated around 1980 by the work of Wiegold-Williamson, and significantly progressed by Liebeck, Praeger and Saxl in 2010. The classification is then used to solve problems in bicrossproduct Hopf algebras and permutation groups.
The classification of the finite primitive permutation groups of rank $3$ was completed in the 1980s and this landmark achievement has found a wide range of applications. In the general transitive setting, a classical result of Higman shows that every finite imprimitive rank $3$ permutation group $G$ has a unique non-trivial block system $\mathcal{B}$ and this provides a natural way to partition the analysis of these groups. Indeed, the induced permutation group $G^{\mathcal{B}}$ is $2$-transitive and one can also show that the action induced on each block in $\mathcal{B}$ is also $2$-transitive (and so both induced groups are either affine or almost simple). In this paper, we make progress towards a classification of the rank $3$ imprimitive groups by studying the case where the induced action of $G$ on a block in $\mathcal{B}$ is of affine type. Our main theorem divides these rank $3$ groups into four classes, which are defined in terms of the kernel of the action of $G$ on $\mathcal{B}$. In particular, we completely determine the rank $3$ semiprimitive groups for which $G^{\mathcal{B}}$ is almost simple, extending recent work of Baykalov, Devillers and Praeger. We also prove that if $G$ is rank $3$ semiprimitive and $G^{\mathcal{B}}$ is affine, then $G$ has a regular normal subgroup which is a special $p$-group for some prime $p$.
We introduce the concept of pseudocover, which is a counterpart of cover, for symmetric graphs. The only known example of pseudocovers of symmetric graphs so far was given by Praeger, Zhou and the first-named author a decade ago, which seems technical and hard to extend to obtain more examples. In this paper, we present a criterion for a symmetric extender of a symmetric graph to be a pseudocover, and then apply it to produce various examples of pseudocovers, including (1) with a single exception, each Praeger-Xu's graph is a pseudocover of a wreath graph; (2) each connected tetravalent symmetric graph with vertex stabilizer of size divisible by $32$ has connected pseudocovers.