Francesca Dalla Volta, Johannes Siemons
We investigate properties of finite transitive permutation groups $(G, Ω)$ in which all proper subgroups of $G$ act intransitively on $Ω.$ In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups.
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We improve the upper bounds (in terms of $n$) in [9] and [13] on the minimal number of elements required to generate a minimally transitive permutation group of degree $n$.
We prove that the base size of a transitive group $G$ with solvable point stabilizer and with trivial solvable radical is not greater than $k$ provided the same statement holds for the group of $G$-induced automorphisms of each nonabelian composition factor of $G$.
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.
Non-solvable $2$-arc-transitive covers of Petersen graphs
本文通过构造 Petersen 图的连通 2-弧传递覆盖并使其变换群不可解，解决了该类覆盖存在性的长期未决问题。
A construction of 2-arc-transitive non-solvable covers of complete graphs
本文提出一种构造完全图2-弧传递非可解覆盖的方法，解决了该类覆盖的存在性问题。
Asymptotic enumeration of minimally transitive permutation groups
该研究证明了极小传递置换群计数的Pyber上界在固定素数幂序列上最优，推导得出顶点传递图和有向图的数量阶，并给出Pyber上界的未发表证明，探讨对McKay–Praeger猜想的启示。
We prove the following result. Let q be a power of an odd prime and let Sp(2m,q) denote the symplectic group of degree 2m over F_q. Then if q=1 mod 4, no solvable subgroup of Sp(2m,q) acts transitively on a complete symplectic spread defined on the underlying vector space of dimension 2m, unless m=1 and q=5. The solvable group Sp(2,3) of order 24 acts transitively on a complete symplectic spread defined on a two-dimensional vector space over F_5. By contrast, when q=3 mod 4, there is a metacyclic group of order 2m(q^m+1) that acts transitively on a complete symplectic spread.
Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given ac…
A complete classification of solvable factors of almost simple groups
本文完成了几乎单群因子分解中可解因子的分类，解决了精确因子分解问题的最后剩余情形，并由此给出一维传递群的新刻画及拟本原置换群中可解传递子群“小”的结论。
Groups Acting Generically Multiply Transitively on Solvable Groups
本文在有限 Morley 秩设定下，完成了群在可解群上泛多重传递作用的分类，证明此类作用等价于一般线性群在代数闭域向量空间上的自然作用，并推广了可定义本原群的分类。
A transitive permutation group of prime degree is doubly transitive or solvable. We give a direct proof of this theorem by Burnside which uses neither S-ring type arguments, nor representation theory.
We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional analytical solvable and transitive Lie algebras of vector fields whose derivative ideal is nilpotent can be adapted to this scheme.
We refer to $d(G)$ as the minimal cardinality of a generating set of a finite group $G$, and say that $G$ is $d$-generated if $d(G)\leq d$. A transitive permutation group $G$ is called $\frac{3}{2}$-transitive if a point stabilizer $G_α$ is nontrivial and its orbits distinct from $\{α\}$ are of the same size. We prove that $d(G)\leq4$ for every primitive $\frac{3}{2}$-transitive permutation group $G$, moreover, $G$ is $2$-generated except for the very particular solvable affine groups that we completely describe. In particular, all finite $2$-transitive and $2$-homogeneous groups are $2$-gener…
The endomorphism rings of permutation modules of $\frac{3}{2}$-transitive permutation groups
本文针对 $\frac{3}{2}$-传递置换群的六类分类结果，通过逐类分析证明了其自然置换模的自同态环均为对称代数。
Finite groups in which strong 4-quasinormality is transitive relation
本文定义并研究强4-拟正规性具有传递性的有限Sq4T-群，证明这类群必可解，并补充4-可置换性相关结论与示例。
We give a sufficient condition for parameter rigidity of actions of solvable Lie groups, by vanishing of (uncountably many) first cohomologies of the orbit foliations. In some cases, we can prove that vanishing of finitely many cohomologies is sufficient. For this purpose we use a rigidity property of quasiisometry. As an application we prove some actions of 2-step solvable Lie groups on mapping tori are parameter rigid. Special cases of these actions are considered in a paper of Matsumoto and Mitsumatsu. We also remark on the relation between transitive locally free actions of solvable Lie gr…
Suppose that $G$ is a finite, transitive, solvable permutation group acting on a set $S$ with $n$ elements. Let $G_0$ be the stabilizer of a point $α\in Ω$. Define the rank of a permutation group, denoted $r(G),$ as the number of distinct orbits of $G_0$ in $S$ (including the trivial orbit $\{α\}$). Huppert \cite{Huppert} and Foulser \cite{Foulser} classified all finite, solvable, permutation groups of rank two and three respectively, and Foulser restricted the rank four groups to a small list of possibilities. This paper completes the classification of all groups of rank less than $5$ by expl…
Arc-transitive maps with coprime Euler characteristic and edge number -- II
作为系列论文第二篇，本文完成不可解自同构群的弧传递地图分类，结合首篇成果完整刻画欧拉示性数与边数互素的弧传递地图，分类过程构建了若干新颖反向地图。
Arc-transitive maps with edge number coprime to the Euler characteristic -- I
本文作为边数与欧拉示性数互素的边传递地图分类系列论文之一，建立分类框架，完成可解自同构群下弧传递地图的分类，揭示边数对欧拉示性数的影响，涉及多种新颖弧正则地图的构造。
In 2011, Fang et al. in (J. Combin. Theory A 118 (2011) 1039-1051) posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency $d$, where either $d\leq 20$ or $d$ is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\geq4$. Even for this problem, it was only solved f…
Base sizes for finite linear groups with solvable stabilisers
本文针对点稳定子可解、可解根基平凡的传递置换群的基大小猜想，证明了基柱同构于PSL_n(q)的殆单群情形下该猜想的强形式，其余典型群将在后续论文处理。
We establish a sharp sufficient condition for groups acting on trees to be highly transitive when the action on the tree is minimal of general type. This gives new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.
We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$. In this paper, we show that such groups are solvable. As an application, we characterise a class of normal edge-transitive Cayley graphs of $G$, which are complete multipartite graphs.
On post-Lie algebras structures coming from simply transitive NIL-affine actions
本文证明了单连通可解 Lie 群在幂零 Lie 群上的单传递 NIL-仿射作用可诱导 post-Lie 代数结构，并在 2 步幂零情形下确立了单传递作用与完备 post-Lie 代数结构的精确对应。
Solvability and Rigidity for Topological Skew Braces
本文研究了有限斜辫子的可解性问题的紧致和局部紧致拓扑类比，探讨可解的加法群是否迫使乘法群可解。主要结论证明在连通局部紧致Hausdorff空间中，若斜辫子的加法群可解，则乘法群也必可解。并通过构造反例说明Hausdorff性、局部紧致性和连通性假设是必要的。在紧致连通Hausdorff且加法群为阿贝尔群的情况下，两种群运算会重合。
Bounding the degree of generic sharp transitivity
本研究在L-群条件下证明了有限Morley秩泛型锐t-传递置换群的次数满足t≤r+2，推进了Borovik-Cherlin猜想，证明基于三个独立的群作用相关关键要素。
A Proof of Gluck's Conjecture
该研究证明了有限可迁置换群与有限可解群的相关不等式，进而解决了自1985年以来悬而未决的格卢克猜想。
Let a finite group $G$ act transitively on a finite set $X$. A subset $S\subseteq G$ is said to be {\it intersecting} if for any $s_1,s_2\in S$, the element $s_1^{-1}s_2$ has a fixed point. The action is said to have the {\it weak Erdős-Ko-Rado} property, if the cardinality of any intersecting set is at most $|G|/|X|$. If, moreover, any maximal intersecting set is a coset of a point stabilizer, the action is said to have the {\it strong Erdős-Ko-Rado} property. In this paper we will investigate the weak and strong Erdős-Ko-Rado property and attempt to classify the groups whose all transitive a…
Structural and rigidity properties of Lie skew braces
本文研究了李歪 brace 的结构和刚性性质，揭示了其在不同李群类别中的存在条件及构造方法。
We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first posit…
Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F\wr\Z, where F is any finite Abelian group; the solvable Baumslag-Soli…
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