Intransitive Cartesian decompositions preserved by innately transitive permutation groups

Identifying Cartesian decompositions preserved by transitive permutation groups

Our aim is to describe the theory of Cartesian decompositions preserved by some member of a large family of finite transitive permutation groups called innatelytransitive groups.

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。

On solvable minimally transitive permutation groups

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.

Intransitive geometries

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.

On intransitive graph-restrictive permutation groups

Let $Γ$ be a finite connected $G$-vertex-transitive graph and let $v$ be a vertex of $Γ$. If the permutation group induced by the action of the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is permutation isomorphic to $L$, then $(Γ,G)$ is said to be locally-$L$. A permutation group $L$ is graph-restrictive if there exists a constant $c(L)$ such that, for every locally-$L$ pair $(Γ,G)$ and a vertex $v$ of $Γ$, the inequality $|G_v|\leq c(L)$ holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.

关于传递群的基大小与最小次数

On the base size and minimal degree of transitive groups

本文证明对任意正数ε，存在传递置换群使基大小与最小次数的乘积至少为n的2-ε次方，并确定具有更小上界的传递与非传递群类。

On the order of vertex-stabilisers in vertex-transitive graphs with local group $C_p\times C_p$ or $C_p \wr C_2$

Let $p$ be a prime and let $L$ be either the intransitive permutation group $C_p\times C_p$ of degree $2p$ or the transitive permutation group $C_p \wr C_2$ of degree $2p$. Let $Γ$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $Γ$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is isomorphic to $L$ then either $|V(Γ)|\geq p|G_v|\log_p\left(|G_v|/2\right)$, or $|V(Γ)|$ is bounded by a constant depending only on $p$, or $Γ$ is a very-well understood graph. This generalises a few recent resul…

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.

Neighbour-transitive codes in Johnson graphs

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset Γsuch that the subgroup G of graph automorphisms leaving Γinvariant is transitive on both the set Γof `codewords' and also the set of `neighbours' of Γ, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the r…

不相容群中的错位

Derangements in intransitive groups

研究不相容群中是否存在错位，证明在特定条件下群确实具有错位，并探讨其与置换族和多项式根的关系。

弱EKR群的刻画及具有指定点稳定子的交集密度

Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers

本文刻画弱EKR群的性质，建立相关准则，还解决公开问题，证明对指定有限群H和整数n≥m，存在对应传递置换群的交集密度为n/m。

Embedding permutation groups into wreath products in product action

The wreath product of two permutation groups G < Sym(Gamma) and H < Sym(Delta) can be considered as a permutation group acting on the set Pi of functions from Delta to Gamma. This action, usually called the product action, of a wreath product plays a very important role in the theory of permutation groups, as several classes of primitive or quasiprimitive groups can be described as subgroups of such wreath products. In addition, subgroups of wreath products in product action arise as automorphism groups of graph products and codes. In this paper we consider subgroups X of full wreath products …

论有限置换群的无冗余基与极小基的基数

On the cardinality of irredundant and minimal bases of finite permutation groups

本文研究有限置换群无冗余基与极小基的基数集合，证明前者为自然数区间而后者未必，并给出使两类群实现给定集合的条件。

Coxeter group actions and limits of hypergeometric series

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set $S$ of 56 distinct translates of a certain function $M$, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised $_9F_8$ hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group $W(E_7)$ on this set. We show that, by removing from $W(E_7)$ a particular generator, we obtain a subgroup that is isomorphic to $W(D_6)$, and that acts intransi…