Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups

关于多循环猜想的一个无穷反例族

An infinite family of counterexamples to the Polycirculant Conjecture

反驳多循环猜想，证明存在无半正则自同构的顶点传递图。采用Chen等人方法构造度为16464的难以捉摸群\(7^6.\mathrm{PSU}_3(3)\)，其是2 - 闭的，此例扩展出多循环猜想无穷反例及相关顶点传递图。

On the order of Borel subgroups of group amalgams and an application to locally-transitive graphs

A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Bo…

具有双正则二面体子群的拟本原群及弧传递双二面体图

Quasiprimitive groups with a biregular dihedral subgroup,and arc-transitive bidihedrants

本文分类了具有双正则二面体子群的拟本原置换群，并据此刻画了自同构群含双正则二面体子群的弧传递图，证明其为正规r-覆盖并分类了底层拟本原或双拟本原图。

Elusive groups of automorphisms of digraphs of small valency

A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.

A classification theorem for boundary 2-transitive automorphism groups of trees

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each vertex contains the alternating group. The outcome of the classification for a fixed tree $T$ is a countable family of groups, all containing two remarkable subgroups: a simple subgroup of index $\leq 8$ and (the semiregular analog of) the universal locally alternating group of Burger-Mozes (with possibly infinite index). We also provide an explicit example sho…

On graph-restrictive permutation groups

Let $Γ$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex of $Γ$ and let $L=G_v^{Γ(v)}$ be the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$. Then $(Γ,G)$ is said to be \emph{locally-$L$}. A transitive permutation group $L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(Γ,G)$ and an arc $(u,v)$ of $Γ$, the inequality $|G_{uv}|\leq c(L)$ holds. Using this terminology, the Weiss Conjecture says that primitive groups are graph-restrictive. We propose a very strong generalisati…

秩 $3$ 的有限半本原置换群

Finite semiprimitive permutation groups of rank $3$

本文给出了非固有传递的秩 $3$ 有限半本原置换群的完整分类，其例子包括某些几乎简单 $2$-传递群的 Schur 覆盖及三个例外小群。

A theory of semiprimitive groups

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 …

Bounds for Finite Semiprimitive Permutation Groups: Order, Base Size, and Minimal Degree

In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.

群作用的线性维度

Linear dimension of group actions

本文研究了群作用的线性维度，通过置换模理论分析了最小维度表示的结构，并探讨了其在密码学中的应用。