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 this paper correspond to three of the six classes. We find that in each case the isomorphism type of the acting group is restricted, and some interesting combinatorial structures are left invariant. We also show how to construct examples of inclusions for each type.
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…
We study $(G,2)$-arc-transitive graphs for innately transitive permutation groups $G$ such that $G$ can be embedded into a wreath product $\symΓ\wr\sy\ell$ acting in product action on $Γ^\ell$. We find two such connected graphs: the first is Sylvester's double six graph with 36 vertices, while the second is a graph with $120^2$ vertices whose automorphism group is $\aut\sp 44$. We prove that under certain conditions no more such graphs exist.
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.
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.
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.
The first main result of this paper is that a finite transitive nonabelian characteristically simple subgroup of a wreath product in product action must lie in the base group of the wreath product. This allows us to characterize nonabelian transitive characteristically simple subgroups $H$ of finite quasiprimitive permutation groups $G$. If the socle of $G$, denoted by $\mbox{soc}(G)$, is nonabelian, then $H$ lies in $\mbox{soc}(G)$. An explicit description is given for the possibilities of $H$ under the condition that $H$ does not contain a nontrivial normal subgroup of $\mbox{soc}(G)$.
Let $(G_n,X_n)$ be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $...\wr G_2\wr G_1$ is topologically finitely generated if and only if the profinite abelian group $\prod_{n\geq 1} G_n/G'_n$ is topologically finitely generated. As a corollary, for a finite transitive group $G$ the minimal number of generators of the wreath power $G\wr...\wr G\wr G$ ($n$ times) is bounded if $G$ is perfect, and grows linearly if $G$ is non-perfect. As a by-product we construct a finitely generate…
We construct the first infinite families of locally arc transitive graphs with the property that the automorphism group has two orbits on vertices and is quasiprimitive on exactly one orbit, of twisted wreath type. This work contributes to Giudici, Li and Praeger's program for the classification of locally arc transitive graphs by showing that the star normal quotient twisted wreath category also contains infinitely many graphs.
Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically) finitely generated if and only if $\prod_{i=1}^{\infty} (G_i/G_i')$ is finitely generated and the growth of the minimal number of generators of $G_i$ is bounded by $ d...n_1...n_{i-1}$ for a constant $d$. Moreover we give a criterion to decide whether $W_\infty$ is positively finitely generated.
We give a partial solution a question of Grigorchuk, Nekrashevych, Sushchanskii and Šunik by giving an algorithm to test whether a finite state element of an infinite iterated (permutational) wreath product $\hat G = \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr >...$ of cyclic groups of order $n$ acts spherically transitively. We can also decide whether two finite state spherically transitive elements of $\hat G$ are conjugate. For general infinite iterated wreath products, an algorithm is presented to determine whether two finite state automorphisms have the same im…
Let $\mathcal{S}$ be a sequence of finite perfect transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated wreath product in product action of the groups in $\mathcal{S}$ is topologically finitely generated, provided that the actions of the groups in $\mathcal{S}$ are not regular. We prove that our bound has the right asymptotic behaviour. We also deduce that other infinitely iterated mixed wreath products of groups in $\mathcal{S}$ are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterat…
We explicitly determine all of the transitive groups of degree p-squared, p a prime, whose Sylow p-subgroup is not the wreath product of two cyclic groups of order p. Furthermore, we provide a general description of the transitive groups of degree p-squared whose Sylow p-subgroup is such a wreath product, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p-squared, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p-squared, and find all nonnormal Cayley grap…
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 …
Let $G$ be an irreducible imprimitive subgroup of $\operatorname{GL}_n(\mathbb{F})$, where $\mathbb{F}$ is a field. Any system of imprimitivity for $G$ can be refined to a nonrefinable system of imprimitivity, and we consider the question of when such a refinement is unique. Examples show that $G$ can have many nonrefinable systems of imprimitivity, and even the number of components is not uniquely determined. We consider the case where $G$ is the wreath product of an irreducible primitive $H \leq \operatorname{GL}_d(\mathbb{F})$ and transitive $K \leq S_k$, where $n = dk$. We show that $G$ ha…
We study the base sizes of finite quasiprimitive permutation groups of twisted wreath type, which are precisely the finite permutation groups with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every permutation group of twisted wreath type is permutation isomorphic to a twisted wreath product $G=T^k{:}P$ acting on its base group $Ω=T^k$, where $T$ is some non-abelian simple group and $P$ is some group acting transitively on $\boldsymbol{k}=\{1,\ldots,k\}$ with $k\geq 2$. We prove that if $G$ is primitive on $Ω$ and $P$ is quasiprimitive on $\boldsymbol{k}$,…
A group is said to be self-similar provided it admits a faithful state-closed representation on some regular $m$-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level of the tree. A standard approach for constructing a transitive self-similar representation of a group has been by way of a single virtual endomorphism of \ the group in question. Recently, it was shown that this approach when applied to the restricted wreath product $% \mathbb{Z}\wr \mathbb{Z}$ could not produce a faithful transitive self-similar representat…
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…
We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group M [X] N is simple. As a permutational product, its most significant property is the following: M [X] N is primitive if and only if M is primitive but not regular, and N is transitive. Despite this remarkable similarity with the wreath product in product action, M [X] N and M Wr N are thoroughly dissimilar. The product provides a general way to build exotic …
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…
Let T_1,..., T_d be homogeneous trees with degrees q_1+1,..., q_d+1>=3, respectively. For each tree, let h:T_j->Z be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of T_1,...,T_d is the graph DL(q_1,...,q_d) consisting of all d-tuples x_1...x_d in T_1x...xT_d with h(x_1)+...+h(x_d)=0, equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If d=2 and q_1=q_2=q then we obtain a Cayley gr…
