Saxl graphs of primitive affine groups with sporadic point stabilisers

零散几乎单群的共轭生成

Conjugate generation of sporadic almost simple groups

本文针对除 Monster 外的所有零散单群及其素数阶自同构，确定了由自同构共轭生成包含原单群之子群所需的最少共轭数。

置换群的Saxl超图

The Saxl hypergraph of a permutation group

该研究提出置换群Saxl超图这一Saxl图的替代推广形式，以大小为基大小的基为边，考察了完全Saxl超图群、带旗支撑环游的本原群及相关猜想推广。

具有散在基座的有限几乎单群中共轭元素的生成

Generation by conjugate elements of finite almost simple groups with a sporadic socle

本文研究散在基座有限几乎单群中自同构的共轭元生成问题，通过特征标表计算细化Guralnick–Saxl估计，证明除Fischer群3-对换情形外α_S(x)≤4，并给出例外群的精确值。

零散单群的贪心基大小

Greedy base sizes for sporadic simple groups

本文研究作用于集合的置换群的基大小问题，针对以零散单群为基座的几乎单本原群，利用自然贪心算法确定了其产生的基的最大大小𝒢(G)，并证明其等于最小基大小b(G)，证实了Cameron猜想。

On base sizes for primitive groups of product type

Let $G \leqslant {\rm Sym}(Ω)$ be a finite permutation group and recall that the base size of $G$ is the minimal size of a subset of $Ω$ with trivial pointwise stabiliser. There is an extensive literature on base sizes for primitive groups, but there are very few results for primitive groups of product type. In this paper, we initiate a systematic study of bases in this setting. Our first main result determines the base size of every product type primitive group of the form $L \wr P \leqslant {\rm Sym}(Ω)$ with soluble point stabilisers, where $Ω= Γ^k$, $L \leqslant {\rm Sym}(Γ)$ and $P \leqsl…

On imprimitive rank 3 permutation groups

A classification is given of rank 3 group actions which are quasiprimitive but not primitive. There are two infinite families and a finite number of individual imprimitive examples. When combined with earlier work of Bannai, Kantor, Liebler, Liebeck and Saxl, this yields a classification of all quasiprimitive rank 3 permutation groups. Our classification is achieved by first classifying imprimitive almost simple permutation groups which induce a 2-transitive action on a block system and for which a block stabiliser acts 2-transitively on the block. We also determine those imprimitive rank 3 pe…

对角型的传播原始群不存在

Spreading primitive groups of diagonal type do not exist

本文证明了对角型的传播原始群不存在，通过展示非阿贝尔有限简单群在特定作用下的性质。

On edge-primitive 3-arc-transitive graphs

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.

有限群的正则数及其他与基相关的不变量

On the regularity number of a finite group and other base-related invariants

本文建立研究有限群无核子群元组正则性的一般方法，确定以交错群或散在群为基柱的几乎单群的精确正则数，并推广若干置换群基大小猜想与结果。

A characterisation of weakly locally projective amalgams related to $A_{16}$ and the sporadic simple groups $M_{24}$ and $He$

A simple undirected graph is weakly $G$-locally projective, for a group of automorphisms $G$, if for each vertex $x$, the stabiliser $G(x)$ induces on the set of vertices adjacent to $x$ a doubly transitive action with socle the projective group $L_{n_x}(q_x)$ for an integer $n_x$ and a prime power $q_x$. It is $G$-locally projective if in addition $G$ is vertex transitive. A theorem of Trofimov reduces the classification of the $G$-locally projective graphs to the case where the distance factors are as in one of the known examples. Although an analogue of Trofimov's result is not yet availabl…

有限置换群的固定元与错排

Fixers and derangements of finite permutation groups

本文对以 PSL_2(q) 为基座的几乎单本原群的大固定元子群进行共轭分类，并验证了 Spiga 猜想的一个特例，同时给出交错群和散在单群情形的相关结果。