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Normal Subgroups of Iterated Wreath Products of Symmetric Groups And Normal subgroups of iterated wreath products of symmetric groups and alternating with symmetric groups Ruslan V. Skuratovskii Abstract Normal subgroups and there properties for finite and infinite iterated wreath products Sn1 ≀ ... ≀ Snm , n,m ∈ N and An ≀ Sn are founded. The special classes of normal subgroups and there orders are investigated. Special classes of normal subgroups are investigated and their generators are found and presented in the form of Kaloujnine tables. Further, the monolith (i.e. the unique minimal normal subgroup) of these wreath products is investigated. Key words: normal subgroups, iterated wreath product of symmetric groups, normal subgroups of wreath product of symmetric group and alternating group. MSC: 20D10, 20F05, 20B05, 20B25, 20B22, 20B07, 20E08, 20E28, 20B35. 1 Introduction This work is the continuation of the previous researches of the author [1, 27]. Also this research clarifies our previous results from [1]. One of the modern brunches of group theory researches is a study of isometry of geometry structures as graph trees and of combinatorial-topological structures [28]. In many cases it is related with studying of groups of automorphisms and normal structures of these groups. Isometry group of Baire space of type (n1, n2, . ) is isomorphic to spherically homogeneous tree with the same type. One of the reasons for the interest in groups acting on trees is that various interesting examples of groups are naturally depicted as groups acting in particular on root trees, arXiv:2108.03752v2 [math.GR] 7 Sep 2021 for example, infinite finitely generated periodic groups. In addition, groups of automor- phisms (isometries) of root trees are still being actively studied due to the fact that they are quite well arranged and immerse in them numerous finitely approximate groups con- taining a number of subgroups with different properties. Thus, the periodic groups of V.I. Sushchanskyy and R. I. Grigorchuk are embedded in the groups of automorphisms of such trees. Thus, it is natural to consider infinitely iterated crown products of finite 1 groups of substitutions. In particular, the main purpose of this work is to describe all nor- mal subgroups of the group of automorphisms of a finite (restricted) regular rooted tree. The paper develops tools for the study of the normal structure, based on the specifics of the wreath product of symmetric groups. A form of commutators of wreath product A ≀ B was briefly considered in [13]. We consider the form of commutator presentation [13] in form of wreath recursion [7,11] and as Kaloujnine tableaux, its properties as a normal subgroup is studied here. We impose our results about of normal subgroups of permutational wreath product [1]. In this paper we continue the research of subgroups of wreath product which was stared in [1, 11, 16, 19, 26] in particular in [11] a commutator subgroup as a normal sub- group of wreath product with non-effective action was considered by us. Despite this, k it was an open question whether the subgroup of ≀ Sni is normal. We generalize the i=1 passive group of this wreath product Sn ≀ Sn to any permutation subgroup B instead of only considering the wreath product of symmetric groups. 2 Preliminaries Let G be a group. The intersection of all non-trivial normal subgroups Mon(G) of G is called the monolith of a group G. If Mon(G) 6=< 1 >, then the group G is called monolithic, and, in this case, Mon(G) is the least non-trivial normal subgroup of G. Definition 1. The unique minimal normal subgroup is called the monolith. According to [10, 15] by tableaux we call all possible infinite tuple of the form: (k−1) a = [ak]k∈N, where ak ∈ F un( M, Gk), k ∈ N. (1) A group of isometries of the generalized Baire space E [6, 8, 9] over a family of sets Mk, k ∈ Z, we denote by I according to [6, 9, 15]. We suppose that for every k ∈ N set Mk is finite. A group of isometries IZp of metric space of integer ring of p-adic numbers is isomor- phic as a permutation group to the iterated wreath product of permutation groups of degree p. Definition 2. We will say that the subset U < G is determined by its k-coordinate sets [U]k, k ∈ N, if this subset consists of the most possible tables a ∈ IZp such that for arbitrary k ∈ N holds [a]k ∈ [U]k. 2 Definition 3. Let A be some set of the tableaux of form (1), then we will call by the k-projection of the set A the set [A]k = |a|k : a ∈ A. We call the set A as k-coordinate set if [A]k = A holds. Definition 4. We denote by A|k a set of all functions ak such that [ε, . , ε, ak,ε...] ∈ [A]k. Recall that such tableau action on element a = [ak]k∈N on element m = (mk)k∈N ∈ M is defined by the rule on element: (k−1) a ak( m) m = (mk )k∈N. The set G of all such tableaux of the form (1) with such action which equipped by such multiplication rule forms a group. The neutral element is the table of form e = [...,ε,ε,...]. Definition 5. We will say that the subset U < G is determined by its k-coordinate sets [U]k, k ∈ N, if this subset consists of the most possible tables a ∈ I such that for arbitrary k ∈ N holds [a]k ∈ [U]k. If U is a splitting subgroup [15] of the group I, which is defined in [15], then k-coordinate subset [U]k are subgroups of the group U, (k) (k) and [Uk] = U T U . Besides, family [U]k, where k ∈ N will generate a uniformly dense subgroup U ′ of a group U. The splitting normal subgroups of the group I were also studied in [9]. Let G be a group acting (from the right) by permutations on a set X and let H be an arbitrary group. Then the (permutational) wreath product H ≀ G is the semidirect product HX ⋋G, where G acts on the direct power HX by the respective permutations of the direct factors. The group Cp or (Cp, X) is equipped with a natural action by the left shift on X = {1,...,p}, p ∈ N. It is well known that the wreath product of permutation groups is associative. The multiplication rule of automorphisms g, h which are presented in the form of wreath recursion [7] g = (g(1), g(2),...,g(d))σg, h = (h(1), h(2),...,h(d))σh, is given by the formula: g · h = (g(1)h(σg (1)), g(2)h(σg (2)),...,g(d)h(σg (d)))σgσh. We define σ as (1, 2,...,p) where p is clear from context. The set X∗ is naturally the vertex set of a regular rooted tree, i.e. a connected graph without cycles and a designated vertex v0 called the root, in which two words are connected by an edge if and only if they are of form v and vx, where v ∈ X∗, x ∈ X. 3 n ∗ ∗ 0 The set X ⊂ X is called the n-th level of the tree X and X = {v0}. Denote the i-th j the vertex of X by vji. Note that the unique vertex vk,i corresponding to each word v in alphabet X. For every automorphism g ∈ AutX∗ and every word v ∈ X∗ we define the ∗ ∗ section (state) g(v) ∈ AutX of g at v by the rule: g(v)(x)= y for x,y ∈ X if and only ∗ k i if g(vx)= g(v)y. The subtree of X induced by the set of vertices ∪i=0X is denoted by X[k]. The restriction of the action of an automorphism g ∈ AutX∗ to the subtree X[l] is denoted by g(v)|X[l] . The restriction g(vij )|X[1] is called the vertex permutation (v.p.) of g at the vertex vij and denoted by gij. We refer to the endomorphism α|v restriction of g in a vertex v [7]. For example, if |X| = 2 then we just have to distinguish active vertices, i.e. the vertices for which α|v is non-trivial. We label every vertex of Xl, 0 ≤ l < k by sign 0 or 1 depending on the action of v.p. on it. The resulting vertex-labeled regular tree is an element of AutX[k]. Let us fix some notations. The commutator of two group elements a and b is denoted by [a, b]= aba−1b−1. We denote by ab = bab−1 the conjugation of a by b. 3 Normal subgroups of wreath product of symmetric groups 3.1 Normal subgroups in Sn ≀ Sm We will notice at once, that further statements and expositions will be also for n > 3, and in the case when two our symmetric groups from wreath product are different. Elements of our wreath product Sn ≀ Sn,n > 5, will present in the form of tableaux, where [a]1 – the active element of the table and a ∈ Sn, [a1, a2, ..., an]2 are the passive elements of the table, ai ∈ Sn, viz.: [a1]1, [a1, a2, . , an]2. The rule of multiplication of elements is described in book of Meldrum [13]. Let us give the following definitions. Definition 6. The minimal number of transpositions in factorization of permutation corresponding to element [ai]j on transposition we will denote by rnk([ai]j). We set rnk(e) = 0. 4 For example, independent transpositions from the Moore generating system [4] for An can be used for factorization of g in product of transpositions.
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