Iterated Wreath Products Research Articles

Wreath Products and Projective System of non-Schurian Association Schemes

A wreath product is a method to construct an association scheme from two association schemes. We determine the automorphism group of a wreath product. We show a known result that a wreath product is Schurian if and only if both components are Schurian, which yields large families of non-Schurian association schemes and non-Schurian $S$-rings. We also study iterated wreath products. Kernel schemes by Martin and Stinson are shown to be iterated wreath products of class-one association schemes. The iterated wreath products give examples of projective systems of non-Schurian association schemes, with an explicit description of primitive idempotents.

On the normalizer of an iterated wreath product

Given a group G and n ≥ 0 , let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn . We prove that the normalizer of W(G, n) in the symmetric group S ( G n ) is equal to M n ⋉ W ( G , n ) , where Mn is isomorphic to Aut ( G ) n . The action of Aut ( G ) n on W(G, n) is recursively described. Communicated by Mark Lewis

Natural Transformations between Induction and Restriction on Iterated Wreath Product of Symmetric Group of Order 2

Let CAn=C[S2≀S2≀⋯≀S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties of An such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups An and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of ⨁m≥0(Am,An)−bimodules. A complete description of the category is an open problem.

Dynamical Belyi maps and arboreal Galois groups

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group A 2 k is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of A 2 k is constructed. It is shown that ( S y l 2 A 2 k ) 2 = S y l 2 ′ A 2 k , k > 2 . It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups C p i , p i ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) of A 2 k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group A 2 k , where k > 2 , the permutation group S 2 k , as well as Sylow p-subgroups of S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B ≀ C n is investigated. An upper bound of the commutator width of permutational wreath product B ≀ C n for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.

Minimal Generating Set and Structure of the Wreath Product for Groups, Commutator of Wreath Product and the Fundamental Group of the Orbit Morse Function

The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{{2^{k}}}}$ is investigated.It is shown that $(Syl_2 A_{2^k})^2 = Syl'_2 A_{2^k}, \, k>2$.It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the commutator width $(cw(G))$ of a wreath product of groups.A recursive presentation of Sylows $2$-subgroups $Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ is introduced. 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 Sylow $p$-subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 is obtained.A commutator width of permutational wreath product $B \wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B \wr C_n$ for an arbitrary group $B$ is found.

Hereditarily just infinite profinite groups with complete Hausdorff dimension spectrum

We prove that the inverse limits of certain iterated wreath products in product action have complete Hausdorff dimension spectrum with respect to their unique maximal filtration of open normal subgroups. Moreover we can produce explicitly subgroups with a specified Hausdorff dimension.

Almost invariance of distributions for random walks on groups

We study the neighborhoods of a typical point $$Z_n$$ visited at n-th step of a random walk, determined by the condition that the transition probabilities stay close to $$\mu ^{*n}(Z_n)$$ . If such neighborhood contains a ball of radius $$C \sqrt{n}$$ , we say that the random walk has almost invariant transition probabilities. We prove that simple random walks on wreath products of $$\mathbb {Z}$$ with finite groups have almost invariant distributions. A weaker version of almost invariance implies a necessary condition of Ozawa’s criterion for the property $$H_\mathrm{FD}$$ . We define and study the radius of almost invariance. We estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n / L(n), where L(n) denotes the drift function of the random walk. We show that the radius of individual almost invariance of a simple random walk on the wreath product of $$\mathbb {Z}^2$$ with a finite group is asymptotically strictly larger than n / L(n). Finally, we show the existence of groups such that the radius of almost invariance is smaller than a given function, but remains unbounded. We also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales.

The Maschke property for the Sylow $p$-sub-groups of the symmetric group $S_{p^n}$

‎‎In this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups like‎: ‎metacyclic $p$-groups‎, ‎$p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groups‎. ‎The main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$‎, ‎i.e‎. ‎on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$‎, ‎where we also prove that a stronger form of the Maschke property holds‎. ‎These results contribute to a future possible classification of all $p$-groups with the Maschke property‎. ‎We apply these results to describe which normal partition subgroups of $P_n$ have a complement‎. ‎In the end we also describe abelian subgroups of $P_n$ of largest size‎.

Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

Let K K be a global function field and let ϕ ( x ) ∈ K [ x ] \phi (x)\in K[x] . For all wandering basepoints b ∈ K b\in K , we show that there is a bound on the size of the elements of the dynamical Zsigmondy set Z ( ϕ , b ) \mathcal {Z}(\phi ,b) that depends only on ϕ \phi , the poles of the b b , and K K . Moreover, when we order b ∈ O K , S b\in \mathcal {O}_{K,S} by height, we show that Z ( ϕ , b ) \mathcal {Z}(\phi ,b) is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of ϕ ( x ) = x d + f \phi (x)=x^d+f is a finite index subgroup of an iterated wreath product of cyclic groups. In particular, since our methods translate to rational function fields in characteristic zero, we establish the inverse Galois problem for these groups via specialization.

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.

Free subgroups of inverse limits of iterated wreath products of non-abelian finite simple groups in primitive actions

Abstract Let 𝒲 = { G i ∣ 1 ≤ i ∈ ℕ } {\mathcal{W}=\{G_{i}\mid 1\leq i\in\mathbb{N}\}} be a set of non-abelian finite simple groups. Set W 1 = G 1 {W_{1}=G_{1}} and choose a faithful transitive primitive W 1 W_{1} -set Δ 1 \varDelta_{1} . Assume that we have already constructed W n - 1 W_{n-1} and chosen a transitive faithful primitive W n - 1 W_{n-1} -set Δ n - 1 \varDelta_{n-1} . The group W n W_{n} is then defined as W n = G n ⁢ wr Δ n - 1 ⁡ W n - 1 {W_{n}=G_{n}\operatorname{wr}_{\varDelta_{n-1}}W_{n-1}} . If W is the inverse limit W = lim ← ⁡ ( W n , ρ n ) {W=}{\varprojlim(W_{n},\rho_{n})} with respect to the natural projections ρ n : W n → W n - 1 {\rho_{n}\colon W_{n}\to W_{n-1}} , we prove that, for each k ≥ 2 k\geq 2 , the set of k-tuples of W that freely generate a free subgroup of rank k is comeagre in W k W^{k} and its complement has Haar measure zero.

Embedding properties of hereditarily just infinite profinite wreath products

We study infinitely iterated wreath products of finite permutation groups w.r.t. product actions. In particular, we prove that, for every non-empty class of finite simple groups X, there exists a finitely generated hereditarily just infinite profinite group W with composition factors in X such that any countably based profinite group with composition factors in X can be embedded into W. Additionally we investigate when infinitely iterated wreath products of finite simple groups w.r.t. product actions are co-Hopfian or non-co-Hopfian.

Invariable generation of iterated wreath products of cyclic groups

Given a sequence $$\{C_i\}_{i \in \mathbb N}$$ of cyclic groups of prime orders, let $$\Gamma _\infty $$ be the inverse limit of the iterated wreath products $$C_m \wr \cdots \wr C_2 \wr C_1$$ . We prove that the profinite group $$\Gamma _\infty $$ is not topologically finitely invariably generated.

On hereditarily just infinite profinite groups obtained via iterated wreath products

Abstract We study a generalisation of the family of non-(virtually pro-p) hereditarily just infinite profinite groups introduced by J. S. Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.

Finite generation of iterated wreath products in product action

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 never regular. We also deduce that certain infinitely iterated wreath products obtained by a mixture of imprimitive and product actions of groups in \({\mathcal{S}}\) are finitely generated. Finally we apply our methods to find explicitly two generators of infinitely iterated wreath products in product action of certain sequences \({\mathcal{S}}\) of 2-generated perfect groups.

LARGE P-GROUPS WITHOUT PROPER SUBGROUPS WITH THE SAME DERIVED LENGTH

AbstractWe construct a subgroup Hd of the iterated wreath product Gd of d copies of the cyclic group of order p with the property that the derived length and the smallest cardinality of a generating set of Hd are equal to d while no proper subgroup of Hd has derived length equal to d. It turns out that the two groups Hd and Gd are the extreme cases of a more general construction that produces a chain Hd=K1<···< Kp−1=Gd of subgroups sharing a common recursive structure. For i ∈ {1,. . .,p−1}, the subgroup Ki has nilpotency class (i+1)d−1.

On some universal construction of minimal topological generating sets for inverse limits of iterated wreath products of non-Abelian finite simple groups

Let $$(H_i)_{i\ge 1}$$(Hi)i?1 be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit $$\ldots \wr H_2\wr H_1$$??H2?H1 of iterated permutational wreath products of the groups $$H_i$$Hi. The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group $$H$$H on a set $$X$$X. We show that the wreath power $$\ldots \wr H\wr H$$??H?H contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet $$X$$X. The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup.

The Automaton Realization of Iterated Wreath Products of Cyclic Groups

We study profinite groups which are infinitely iterated wreath products W ∞ = …≀C n 2 ≀C n 1 of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the group W ∞ by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton.