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.
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.
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.
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 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…
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…
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…
We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.
We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated)…
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…
For a non-cyclic finite group $X$ let $σ(X)$ be the least number of proper subgroups of $X$ whose union is $X$. Precise formulas or estimates are given for $σ(S \wr C_{m})$ for certain nonabelian finite simple groups $S$ where $C_m$ is a cyclic group of order $m$.
The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the commutator width of such groups is 1 and we research some properties of its commutator subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and …
Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian group G. We find that Z wr Z is undistorted in F but is at least exponentially distorted in G.
This paper investigate bounds of the commutator width \cite {Mur} of a wreath product of two groups. The commutator width of direct limit of wreath product of cyclic groups are found. For given a permutational wreath product sequence of cyclic groups we investigate its commutator width and some properties of its commutator subgroup. It was proven that the commutator width of an arbitrary element of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. As a corollary, it is shown that the commutator width of Sylows $p$-subgroups of symmetric and alternating groups $p…
For each finite classical group $G$, we classify the subgroups of $G$ which act transitively on a $G$-invariant set of subspaces of the natural module, where the subspaces are either totally isotropic or nondegenerate. Our proof uses the classification of the maximal factorisations of almost simple groups. As a first application of these results we classify all point-transitive subgroups of automorphisms of finite thick generalised quadrangles.
Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not…
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…
A permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular. Given nontrivial finite transitive permutation groups $L_1$ and $L_2$ with $L_1$ not semiprimitive, we construct an infinite family of rank two amalgams of permutation type $[L_1,L_2]$ and Borel subgroups of strictly increasing order. As an application, we show that there is no bound on the order of edge-stabilisers in locally $[L_1,L_2]$ graphs. We also consider the corresponding question for amalgams of rank $k\geq 3$. We completely resolve this by showing that the order of the Bo…
In the first paper in this series we estimated the probability that a random permutation $π\in\mathcal{S}_n$ has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that $π$ has $m$ disjoint fixed sets of prescribed sizes $k_1,\dots,k_m$, where $k_1+\cdots+k_m=n$. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than $\mathcal{S}_n$ or $\mathcal{A}_n$. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estim…
For a finite group $G$ and an inverse-closed generating set $C$ of $G$, let $Aut(G;C)$ consist of those automorphisms of $G$ which leave $C$ invariant. We define an $Aut(G;C)$-invariant normal subgroup $Φ(G;C)$ of $G$ which has the property that, for any $Aut(G;C)$-invariant normal set of generators for $G$, if we remove from it all the elements of $Φ(G;C)$, then the remaining set is still an $Aut(G;C)$-invariant normal generating set for $G$. The subgroup $Φ(G;C)$ contains the Frattini subgroup $Φ(G)$ but the inclusion may be proper. The Cayley graph $Cay(G,C)$ is normal edge-transitive if $A…
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.
In this paper we study $G$-arc-transitive graphs $Δ$ where the permutation group $G_x^{Δ(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $Δ(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $Δ$, if $(x,y)$ is an arc of $Δ$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $Δ(x)$ and $Δ(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-l…
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.
The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non) existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht, Graver …
We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characte…
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…
Generators and defining relations for wreath products of groups are given. Under some condition (conormality of the generators) they are minimal. In particular, it is just the case for the Sylow subgroups of the symmetric groups.
The covering number of a finite group $G$, denoted $σ(G)$, is the smallest positive integer $k$ such that $G$ is a union of $k$ proper subgroups. We calculate $σ(G)$ for a family of primitive groups $G$ with a unique minimal normal subgroup $N$, isomorphic to $A_n^m$ with $n$ divisible by $6$ and $G/N$ cyclic. This is a generalization of a result of E. Swartz concerning the symmetric groups. We also prove an asymptotic result concerning pairwise generation.
This paper studies whether there are distance-transitive graphs arising from the coset actions of $G_2(q)$ on the subfield subgroup $G_2(\sqrt{q})$ or $G_2(q)$ on the Ree subgroup $^2G_2(q)$. It is found that there are no such graphs, even if the groups are extended by outer automorphisms of $G_2(q)$.
Wreath products of non-discrete locally compact groups are usually not locally compact groups, nor even topological groups. We introduce a natural extension of the wreath product construction to the setting of locally compact groups. As an application, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without nontrivial compact normal subgroups.
Groups with a large $p$-subgroup, $p$ a prime, include almost all of the groups of Lie type in characteristic $p$ and so the study of such groups adds to our understanding of the finite simple groups. In this article we study a special class of such groups which appear as wreath product cases of the Local Structure Theorem.
A connected graph $Γ=(V,E)$ of valency at least $3$ is called a basic $2$-arc-transitive graph if its full automorphism group has a subgroup $G$ with the following properties: (i) $G$ acts transitively on the set of $2$-arcs of $Γ$, and (ii) every minimal normal subgroup of $G$ has at most two orbits on $V$. In her papers [17,18], Praeger proved a connected $2$-arc-transitive graph of valency at least $3$ is a normal cover of some basic $2$-arc-transitive graph, and characterized the group-theoretic structures for basic $2$-arc-transitive graphs. Based on Praeger's theorems on $2$-arc-transiti…
For a finitely generated regular wreath product, the metric is known, but its computation can be an NP-complete problem. Also, it is not known for the nonregular case. In this article, a metric estimate is defined for regular wreath products which can be computed in polynomial time, based on the metrics of the factors. This estimate is then used to study the distortion of some natural subgroups of a wreath product. Finally, the metric estimate is generalized to the nonregular case.
Let $K$ be field of characteristic 2 and let $G$ be a finite non-abelian 2-group with the cyclic derived subgroup $G'$, and there exists a central element $z$ of order 2 in $Z(G) \backslash G'$. We prove that the unit group of the group algebra $KG$ possesses a section isomorphic to the wreath product of a group of order 2 with the derived subgroup of the group $G$, giving for such groups a positive answer to the question of A. Shalev.
In this paper, we classify finite quasiprimitive permutation groups with a metacyclic transitive subgroup, solving a problem initiated by Wielandt in 1949. It also involves the classification of factorizations of almost simple groups with a metacyclic factor.
We determine the semi-regular subgroups of the 2-transitive permutation groups PGL(2,n), PSL(2,n), PGU(3,n), PSU(3,n), Sz(n) and Ree(n) with n a suitable power of a prime number p.
We prove an upper bound for the number of cyclic transitive subgroups in a finite permutation group and clarify the structure of the groups for which this bound becomes sharp. We also give an application in the theory of number fields.
In this paper we calculate the character table of a sharply $5$-transitive subgroup of ${\rm Alt}(12)$, and of a sharply $4$-transitive subgroup of ${\rm Alt}(11)$. Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.
Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_…
We show that any group $G$ is contained in some sharply 2-transitive group $\mathcal{G}$ without a non-trivial abelian normal subgroup. This answers a long-standing open question. The involutions in the groups $\mathcal{G}$ that we construct have no fixed points.
We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.
If $Γ$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sym(n) (other than Sym(n) and the alternating group Alt(n)), then the rank of $Γ$ is at most $n/2+1$, with finitely many exceptions (which are classified). It is conjectured that only the symmetric group has to be excluded.
We give a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under taking the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing met…
An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to c…
