Robert W. Baddeley, Cheryl E. Praeger, Csaba Schneider
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.
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We study Cartesian decompositions of sets that are acted upon intransitively by innately transitive permutation groups. We prove that such groups have at most three orbits on such a decomposition. A consequence of this result is that if $G$ is an innately transitive subgroup of a wreath product in product action then the natural projection of $G$ into the top group has at most 2 orbits.
A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be pre…
A permutation group is innately transitive if it has a transitive minimal normal subgroup, and this subgroup is called a plinth. In this paper we study three special types of inclusions of innately transitive permutation groups in wreath products in product action. This is achieved by studying the natural Cartesian decomposition of the underlying set that correspond to the product action of a wreath product. Previously we identified six classes of Cartesian decompositions that can be acted upon transitively by an innately transitive group with a non-abelian plinth. The inclusions studied in th…
Transitive path decompositions of Cartesian products of complete graphs
本文研究完全图笛卡尔积 $K_n \Box K_n$ 的传递路径分解，当 $n$ 为奇素数时构造了路径长度相对于顶点数较大的分解，支持了 Gallai 猜想和 Ringel 猜想的扩展版本。
A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of `Cartesian decompositions' of the permuted set, relating them to certain `Cartesian systemsof subgroups'. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.
A transitive decomposition of a graph is a partition of the edge or arc set giving a set of subgraphs which are preserved and permuted transitively by a group of automorphisms of the graph. In this paper we give some background to the study of transitive decompositions and highlight a connection with partial linear spaces. We then describe a simple method for constructing transitive decompositions using graph quotients, and we show how this may be used in an application to modular origami.
Let $G$ be a permutation group on a finite set $Ω$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ having the same orbits as $G$ on the $k$-th Cartesian power $Ω^k$ of $Ω$. A group $G$ is called $\frac{3}{2}$-transitive if its transitive and the orbits of a point stabilizer $G_α$ on the set $Ω\setminus\{α\}$ are of the same size greater than one. We prove that the $2$-closure $G^{(2)}$ of a $\frac{3}{2}$-transitive permutation group $G$ can be found in polynomial time in size of $Ω$. In addition, if the group $G$ is not $2$-transitive, then for ev…
The classification of flag-transitive generalised quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalised quadrangles are also point-primitive (up to point-line duality), it is likewise natural to seek a classification of the point-primitive examples. Working towards this aim, we are led to investigate generalised quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of…
This paper is about the structure of infinite primitive permutation groups and totally disconnected locally compact groups ("tdlc groups'"). The permutation groups we investigate are subdegree-finite (i.e. all orbits of point stabilisers are finite). Automorphism groups of connected, locally finite graphs are examples of subdegree-finite permutation groups. The tdlc groups we investigate all have a maximal subgroup that is compact and open. Tdlc groups with few open subgroups (recently studied by Pierre-Emmanuel Caprace and Timothée Marquis) are examples of such groups. We prove a classificati…
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