Conjugacy in finite state wreath powers of finite permutation groups

It is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are conjugated in the finite state wreath power of this group. Counter-examples for non-periodic elements are given.

Збережено в:

Бібліографічні деталі
Дата:	2019
Автори:	Oliynyk, A., Russyev, A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2019
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/188422
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Conjugacy in finite state wreath powers of finite permutation groups / A. Oliynyk, A. Russyev // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 58–69. — Бібліогр.: 6 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-188422
record_format	dspace
spelling	irk-123456789-1884222023-03-01T01:27:01Z Conjugacy in finite state wreath powers of finite permutation groups Oliynyk, A. Russyev, A. It is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are conjugated in the finite state wreath power of this group. Counter-examples for non-periodic elements are given. 2019 Article Conjugacy in finite state wreath powers of finite permutation groups / A. Oliynyk, A. Russyev // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 58–69. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC: 20E08, 20E06, 20F65. http://dspace.nbuv.gov.ua/handle/123456789/188422 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	It is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are conjugated in the finite state wreath power of this group. Counter-examples for non-periodic elements are given.
format	Article
author	Oliynyk, A. Russyev, A.
spellingShingle	Oliynyk, A. Russyev, A. Conjugacy in finite state wreath powers of finite permutation groups Algebra and Discrete Mathematics
author_facet	Oliynyk, A. Russyev, A.
author_sort	Oliynyk, A.
title	Conjugacy in finite state wreath powers of finite permutation groups
title_short	Conjugacy in finite state wreath powers of finite permutation groups
title_full	Conjugacy in finite state wreath powers of finite permutation groups
title_fullStr	Conjugacy in finite state wreath powers of finite permutation groups
title_full_unstemmed	Conjugacy in finite state wreath powers of finite permutation groups
title_sort	conjugacy in finite state wreath powers of finite permutation groups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2019
url	http://dspace.nbuv.gov.ua/handle/123456789/188422
citation_txt	Conjugacy in finite state wreath powers of finite permutation groups / A. Oliynyk, A. Russyev // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 1. — С. 58–69. — Бібліогр.: 6 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT oliynyka conjugacyinfinitestatewreathpowersoffinitepermutationgroups AT russyeva conjugacyinfinitestatewreathpowersoffinitepermutationgroups
first_indexed	2025-07-16T10:27:25Z
last_indexed	2025-07-16T10:27:25Z
_version_	1837798938322665472
fulltext	“adm-n1” — 2019/3/22 — 12:03 — page 58 — #66 Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 27 (2019). Number 1, pp. 58–69 c© Journal “Algebra and Discrete Mathematics” Conjugacy in finite state wreath powers of finite permutation groups Andriy Oliynyk and Andriy Russyev Communicated by A. P. Petravchuk Abstract. It is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are con- jugated in the finite state wreath power of this group. Counter- examples for non-periodic elements are given. 1. Introduction The conjugacy classes in the full automorphism group of a regular rooted tree are described in [1]. But for its subgroup of finite state automor- phisms the corresponding description is a challenging task. In particular, there exist finite state level-transitive automorphisms (and therefore, con- jugated in the full automorphism group) which are not conjugates in the finite state subgroup [2]. Deep results about conjugation of some special finite state automorphisms were obtained in [3] and [4]. The most natural way to introduce finite state automorphisms uses automata theory. But regarding our purposes we choose a language of infinite wreath products. The full automorphism group of m-regular rooted tree is the infinite wreath power of the symmetric group of degree m. If we restrict ourselves to some subgroup (or even subsemigroup) G of this symmetric group we naturally obtain the infinite and finite state wreath powers of G (cf. [5, 6]). 2010 MSC: 20E08, 20E06, 20F65. Key words and phrases: permutation group, wreath power, automorphism of a rooted tree, conjugacy. “adm-n1” — 2019/3/22 — 12:03 — page 59 — #67 A. Oliynyk, A. Russyev 59 The work is organized as follows. In Section 2 we recall the finite state wreath power of a finite permutation group. In Section 3 we prove the main result of the paper. Namely, if periodic finite state elements of the infinite wreath power of a finite permutation group are conjugated in this wreath power then they are conjugated in the finite state wreath power of a given permutation group. In Section 4 we show how to construct non-periodic finite state elements conjugated in the wreath power but not conjugated in the finite state wreath power of a given permutation group. For all other definitions used in the paper one can refer to [2]. 2. Finite state wreath power Let A be a finite set of cardinality m > 2. Consider a finite group G acting faithfully on the set A. In other words, the permutation group (G,A) is a subgroup of the symmetric group Sym(A). In the sequel we assume that the groups act on the sets from the right and denote by ag the result of the action of a group element g on a point a. Denote by W∞(G,A) the infinitely iterated wreath product of (G,A). The group W∞(G,A) consists of permutations of the infinite cartesian product X∞ given by infinite sequences of the form g = [ g1, g2(x1), . . . , gn(x1, . . . , xn−1), . . . ], (1) where g1 ∈ G, g2(x1) : A → G, . . . , gn(x1, . . . , xn−1) : A n−1 → G, . . . For each n > 1 we call gn the n-th term of g and denote it by [g]n. An element g acts on a point ā = (a1, a2, . . . , an, . . .) ∈ A∞ by the rule āg = (ag11 , a g2(a1) 2 , . . . , agn(a1,...,an−1) n , . . .). Let g = [g1, g2(x1), g3(x1, x2), . . .] ∈ W∞(G,A) and ā = (a1, . . . , an) ∈ An for some n > 1. Define an element rest(g, ā) ∈ W∞(G,A) as rest(g, ā) = [h1, h2(x1), h3(x1, x2), . . .], where h1 = gn+1(a1, . . . , an), h2(x1) = gn+2(a1, . . . , an, x1), h3(x1, x2) = gn+3(a1, . . . , an, x1, x2), . . . “adm-n1” — 2019/3/22 — 12:03 — page 60 — #68 60 Conjugacy in finite state wreath powers The tuple rest(g, ā) has the form (1) as well and therefore belongs to the group W∞(G,A). The element rest(g, ā) is called the state of g at ā. Also we consider g as a state of itself. Define the set Q(g) = {rest(g, ā) : ā ∈ An, n > 1} ∪ {g} of all states of g. In particular, for the identity element e ∈ W∞(G,A) we get the equality Q(e) = {e}. The following lemma is directly verified. Lemma 1. Let g, h ∈ W∞(G,A). Then Q(gh) ⊆ Q(g) · Q(h), Q(g−1) = Q(g)−1. If h ∈ Q(g) then Q(h) ⊆ Q(g). Let FW∞(G,A) = {g ∈ W∞(G,A) : \|Q(g)\| < ∞}. Lemma 1 implies that this set form a subgroup of W∞(G,A). Definition 1. The group FW∞(G,A) is called the finite state wreath power of the permutation group (G,A). The group FW∞(G,A) is countable while W∞(G,A) is not. Another useful remark is that both permutation groups (W∞(G,A),A∞) and (FW∞(G,A), A∞) split into the wreath product of (G,A) and itself, i.e. (W∞(G,A), A∞) = (G,A) ≀ (W∞(G,A), A∞) (2) and (FW∞(G,A), A∞) = (G,A) ≀ (FW∞(G,A), A∞). (3) This allows us to present an element g = [g1, g2(x1), g3(x1, x2), . . .] ∈ W∞(G,A) in the form g = [g1; rest(g, a), a ∈ A]. (4) 3. Conjugation of periodic elements It is convenient for us to identify the set A with the set {1, . . . ,m}. We need some additional notation here. Let elements g ∈ G and a ∈ A be fixed. Consider the cyclic decomposition of g as a permutation on A. The length of the cycle containing a will be denoted by l(g, a). Then ag l(g,a) = a “adm-n1” — 2019/3/22 — 12:03 — page 61 — #69 A. Oliynyk, A. Russyev 61 and l(g, a) is the smallest integer satisfying this equality. The minimal element of this cycle we denote by s(g, a). Note, that l(g, s(g, a)) = l(g, a). The smallest integer d > 0 such that ag d = s(g, a) will be denoted by d(g, a). Then 0 6 d(g, a) 6 l(g, a)− 1. We have the following Lemma 2. Let u, v and h be elements of G such that u = h−1vh. Then for arbitrary a ∈ A the equality l(u, ah) = l(v, a) holds. Proof. From the definition of l(v, a) we have av l(v,a) = a. Thus (ah)u l(v,a) = (ah)(h −1vh)l(v,a) = (ah)h −1vl(v,a)h = (av l(v,a) )h = ah. Hence, l(u, ah) 6 l(v, a). The inequality l(u, ah) > l(v, a) is proved analo- gously. The rules of multiplication and taking inverses in wreath products imply that for arbitrary u = [u1, . . .], v = [v1, . . .] ∈ W∞(G,A) and a ∈ A one have the equalities: rest(uv, a) = rest(u, a)rest(v, au1) and rest(u−1, a) = rest(u, au −1 1 )−1. Then we can prove Lemma 3. Let u = [u1, . . .], v = [v1, . . .] and h = [h1, . . .] be elements of the group W∞(G,A) such that u = h−1vh. Then for arbitrary a ∈ A the equality rest(ul(u1,ah1 ), ah1) = (rest(h, a))−1rest(vl(v1,a), a)rest(h, a) holds. Proof. Note, that u1 = h−1 1 v1h1. By Lemma 2 we have equalities ul(u1,ah1 ) = ul(v1,a) = h−1vl(v1,a)h. Hence, rest(ul(u1,ah1 ), ah1) = rest(h−1vl(v1,a)h, ah1) = rest(h−1, ah1)rest(vl(v1,a), a)rest(h, av l(v1,a) 1 ) = (rest(h, a))−1rest(vl(v1,a), a)rest(h, a). “adm-n1” — 2019/3/22 — 12:03 — page 62 — #70 62 Conjugacy in finite state wreath powers Theorem 1. Arbitrary elements of finite order of the group FW∞(G,A), conjugated in the group W∞(G,A), are conjugated in the group FW∞(G,A) as well. Proof. Denote by M the set of all pairs (u, v) ∈ FW∞(G,A)×FW∞(G,A) such that elements u and v have finite order and are conjugated in the group W∞(G,A). For each pair θ = (u, v) ∈ M let us fix an element Ψ(θ) ∈ W∞(G,A) such that the equality u = (Ψ(θ))−1vΨ(θ) holds. The correspondence θ 7→ Ψ(θ) may be regarded as a mapping Ψ : M → W∞(G,A). Now we proceed as follows. Using Ψ we construct new mappings Ψ∗ : M → W∞(G,A) and Φ : M ×A → W∞(G,A). Then we show that for each pair θ = (u, v) ∈ M the equality u = (Ψ∗(θ)) −1vΨ∗(θ) holds. Finally, we prove that indeed Ψ∗(M) ⊂ FW∞(G,A). Then the statement of the theorem follows. For a pair θ = (u, v) ∈ M , where u = [u1, u2(x1), . . .] and u = [v1, v2(x1), . . .], we will use the notation Ψ(θ) = [h1, h2(x1), . . .]. Step 1. Let us define mappings Ψ∗ and Φ. It is sufficient to check that recursive equalities Ψ∗(θ) = [h1; rest(v d(v1,a), a)Φ(θ, a) ( rest(ud(v1,a), ah1) ) −1 , a ∈ A], (5) Φ(θ, a) = Ψ∗ ( rest ( ul(u1,ah1 ), (s(v1, a)) h1 ) , rest ( vl(v1,a), s(v1, a) ) ) , a ∈ A (6) correctly define required mappings Ψ∗ and Φ. First of all, from (5) we have [Ψ∗(θ)]1 = [Ψ(θ)]1 and hence the term [Ψ∗(θ)]1 is well-defined. To define other terms we need Φ(θ, a), a ∈ A. Then, by Lemma 3, for each a ∈ A the pair ( rest ( ul(u1,ah1 ), (s(v1, a)) h1 ) , rest ( vl(v1,a), s(v1, a) ) ) belongs to M . Hence, equality (6) defines the first term of Φ(θ, a), a ∈ A. Again looking at (5), we obtain the second term of Ψ∗(θ) and so on. Inductively, for arbitrary k > 1, having defined the kth term of Ψ∗(θ) by (5), we define the kth term of Φ(θ) by (6) and this gives us a possibility to define the (k + 1)th term of Ψ∗(θ) by (5). “adm-n1” — 2019/3/22 — 12:03 — page 63 — #71 A. Oliynyk, A. Russyev 63 Note that for every a ∈ A the equality Φ(θ, av1) = Φ(θ, a) holds. Step 2. Let us prove the equality u = (Ψ∗(θ)) −1vΨ∗(θ), where θ = (u, v) ∈ M . We will prove by induction on k the equality [u]k = [(Ψ∗(θ)) −1vΨ∗(θ)]k. Since [Ψ∗(θ)]1 = [Ψ(θ)]1 and [u]1 = [(Ψ(θ))−1vΨ(θ)]1 we obtain the required statement for k = 1. Assume that for the (k − 1)th terms the equality is proved. Proceed with the kth ones. Fix an element a ∈ A. Denote by g the state of (Ψ∗(θ)) −1vΨ∗(θ) at ah1 . It is sufficient to check the equality [rest(u, ah1)]k = [g]k. For g we have the equalities: g = rest ( (Ψ∗(θ)) −1vΨ∗(θ), a h1 ) = rest ( (Ψ∗(θ)) −1, ah1 ) rest(v, a)rest ( Ψ∗(θ), a v1 ) = (rest(Ψ∗(θ), a)) −1rest(v, a)rest(Ψ∗(θ), a v1) = rest(ud(v1,a), ah1)(Φ(θ, a))−1(rest(vd(v1,a), a))−1rest(v, a) · rest(vd(v1,a v1 ), av1)Φ(θ, av1)(rest(ud(v1,a v1 ), av1h1))−1 = rest(ud(v1,a), ah1)(Φ(θ, a))−1(rest(vd(v1,a), a))−1 · rest(vd(v1,a v1 )+1, a)Φ(θ, av1)(rest(ud(v1,a v1 ), av1h1))−1. There are two possibilities: s(v1, a) = a or s(v1, a) 6= a. Consider these cases. 1) Let s(v1, a) = a. Then d(v1, a) = 0 and d(v1, a v1) = l(v1, a) − 1. This implies rest(ud(v1,a), ah1) = e and rest(vd(v1,a), a) = e. Lemma 2 and equality (6) then implies Φ(θ, a) = Ψ∗(rest(u l(v1,a), ah1), rest(vl(v1,a), a)). Then, in view of the inductive hypothesis, the equalities follow: [g]k = [(Φ(θ, a))−1rest(vl(v1,a), a)Φ(θ, av1)(rest(ul(v1,a)−1, av1h1))−1]k = [rest(ul(v1,a), ah1)(rest(ul(v1,a)−1, ah1u1))−1]k = [rest(u, ah1)rest(ul(v1,a)−1, ah1u1)(rest(ul(v1,a)−1, ah1u1))−1]k = [rest(u, ah1)]k. “adm-n1” — 2019/3/22 — 12:03 — page 64 — #72 64 Conjugacy in finite state wreath powers 2) Let s(v1, a) 6= a. Then d(v1, a v1) = d(v1, a)− 1. For g now we have: g = rest(ud(v1,a), ah1)(Φ(θ, a))−1(rest(vd(v1,a), a))−1 · rest(vd(v1,a v1 )+1, a)Φ(θ, av1)(rest(ud(v1,a v1 ), av1h1))−1 = rest(ud(v1,a), ah1)(rest(ud(v1,a)−1, ah1u1))−1 = rest(u, ah1)rest(ud(v1,a)−1, ah1u1)(rest(ud(v1,a)−1, ah1u1))−1 = rest(u, ah1). In both cases we obtained the equality [rest(u, ah1)]k = [g]k. Hence, our statement is true for the kth terms. Step 3. Let us check the inclusion Ψ∗(M) ⊂ FW∞(G,A). Denote by Mk the subset of all pairs (u, v) ∈ M such that the orders of u and v equal k. These subsets are pairwise disjoint and M = ∞ ⋃ k=1 Mk. Let us prove by induction on k that Ψ∗(Mk) ⊂ FW∞(G,A). In case k = 1 we have M1 = {(e, e)}. Since rest(e, a) = e, a ∈ A, equalities (5) and (6) imply rest(Ψ∗(e, e), a) = Ψ∗(e, e), a ∈ A. Hence, g ∈ FW∞(G,A). Suppose that Ψ∗(Mi) ⊂ FW∞(G,A) for all i < k. We are going to prove the inclusion Ψ∗(Mk) ⊂ FW∞(G,A). If the set Mk is empty then the statement is true. Let θ = (u, v) be a pair belonging to the set Mk. We have to show that \|Q(Ψ∗(θ))\| < ∞. For an element g ∈ W∞(G,A) its set of stable states is defined as SQ(g) = {rest(g, ā) : rest(g2, ā) = (rest(g, ā))2, ā ∈ An, n > 1} ∪ {g}. Since u, v ∈ FW∞(G,A) the set Q1 = Ψ∗ ( ( SQ(u)× SQ(v) ) ∩Mk ) is finite. In particular, Ψ∗ ( (u, v) ) ∈ Q1. We are going to show that Q1 ⊂ FW∞(G,A). “adm-n1” — 2019/3/22 — 12:03 — page 65 — #73 A. Oliynyk, A. Russyev 65 Fix arbitrary g ∈ Q1. Then g = Ψ∗ ( (ũ, ṽ) ) for some ũ ∈ SQ(u), ṽ ∈ SQ(v) such that (ũ, ṽ) ∈ Mk. Let a ∈ A. Denote l([ṽ]1, a) by ℓ. Two possible cases arise. Case 1: ℓ = 1. Then s([ṽ]1, a) = a and d([ṽ]1, a) = 0. By equalities (5) and (6) we now obtain rest(g, a) = rest ( Ψ∗((ũ, ṽ)), a ) = Φ ( (ũ, ṽ), a ) = Ψ∗ ( rest ( ũ, a[Ψ((ũ,ṽ))]1 ) , rest(ṽ, a) ) . The latter element belongs to Q1. Hence, in this case rest(g, a) ∈ Q1. Case 2: ℓ > 1. By Lemma 2 the equality l([ũ]1, a [Ψ((ũ,ṽ))]1)= l([ṽ]1, a)=ℓ holds. By (6) we have Φ ( (ũ, ṽ), a ) = Ψ∗ ( rest ( ũℓ, (s([ṽ]1, a)) [Ψ((ũ,ṽ))]1 ) , rest ( ṽℓ, s([ṽ]1, a) ) ) . (7) Since cyclic decompositions of both [ũ]1 and [ṽ]1 contain a cycle of length ℓ, the number ℓ divides the orders of both ũ and ṽ. It implies that the orders of arguments of Ψ∗ in (7) are strictly less then k. Indeed, they are states of the ℓth powers of elements ũ and ṽ correspondingly at elements belonging to cycles of length ℓ. Applying the inductive hypothesis we get Φ((ũ, ṽ), a) ∈ FW∞(G,A). Now from the definition of Ψ∗ we obtain that the state rest(g, a) = rest ( Ψ∗((ũ, ṽ)), a ) belongs to FW∞(G,A) as the product of elements from FW∞(G,A). Thus, the state rest(g, a) belongs to the finite set Q1 or the set Q(rest(g, a)) is finite. Let Q2 = {rest(g, a) : g ∈ Q1, a ∈ A} ∩ FW∞(G,A) and Q3 = ⋃ h∈Q2 Q(h). Since sets Q1 and A are finite, the set Q2 is finite. Being a union of finite number of finite sets, the set Q3 is finite as well. Then, using the definition of the state we obtain Q(g) ⊂ Q1 ∪Q3. Therefore, g ∈ FW∞(G,A). The proof is complete. “adm-n1” — 2019/3/22 — 12:03 — page 66 — #74 66 Conjugacy in finite state wreath powers Observe that rewriting mapping Ψ∗ constructed in the proof of the- orem 1 may be defined on the set of all pairs of conjugated elements of W∞(G,A). Additional conditions on such elements were used only to prove that the image of Ψ∗ belongs to the finite state wreath power of (G,A). It would be interesting to examine this image in general case. 4. Non-conjugated elements of infinite order Let us show how to construct two elements of the group FW∞(G,A) which are conjugated in the group W∞(G,A) but are not conjugated in the group FW∞(G,A). Let g be a non-identity element of the group G. If n is the order of the element g and p is a prime divisor of n then the element g∗ = gn/p has order p and as a permutation on A is a product of independent cycles of length p. Without loss of generality we assume that g∗ has no fixed points. We will identify the set A with the set {0, . . . ,m− 1} in such a way that for some k > 1 the permutation g∗ will be expressed in the form g∗ = (0, . . . , p− 1)(p, . . . , 2p− 1) · · · ((k − 1)p, . . . , kp− 1). Let us consider the set Ap = {0, . . . , p − 1}, the cyclic group Gp = 〈σ〉 generated by the permutation σ = (0, . . . , p − 1) and the mapping c : Gp → G that maps an element h ∈ Gp to the permutation acting on the set {0, . . . , kp− 1} by the rule x 7→ (x mod p)h + [x/p] · p and trivially on other elements of the set A. In other words the mapping c duplicates action on the set Ap onto the sets {p, . . . , 2p− 1}, . . . , {(k − 1)p, . . . , kp− 1}. Using mapping c one can transform any element u ∈ W∞(Gp, Ap) into an element u(k) ∈ W∞(G,A) by the equality [u(k)]n(x1, . . . , xn−1) =      c ( [u]n(x1 mod p, . . . , xn−1 mod p) ) , 0 6 x1, . . . , xn−1 < kp, e, otherwise. Denote by f the function that for any u ∈ W∞(Gp, Ap) computes u(k) ∈ W∞(G,A). The function f is well-defined. Lemma 4. If u ∈ FW∞(Gp, Ap) then u(k) ∈ FW∞(G,A). Proof. If u ∈ W∞(Gp, Ap) then the value of [u]n equals to some power of σ. By definition of the transformation the value of [u(k)]n equals to the same power of g∗ or e depending on the arguments. Thus u(k) ∈ W∞(G,A). “adm-n1” — 2019/3/22 — 12:03 — page 67 — #75 A. Oliynyk, A. Russyev 67 Denote by Akp the set {0, . . . , kp − 1} and denote by ā mod p the element (a1 mod p, a2 mod p, . . . , an mod p) ∈ An p for ā = (a1, a2, . . . , an) ∈ An. We are going to prove for u ∈ W∞(Gp, Ap) that rest(f(u), ā) = { f(rest(u, ā mod p)), ā ∈ An kp, n > 1, e, otherwise. (8) For ā = (a1, . . . , an) ∈ An and n > 1 we have [rest(f(u), ā)]m(x1, . . . , xm−1) = [f(u)]n+m(a1, . . . , an, x1, . . . , xm−1). If ā /∈ An kp then [rest(f(u), ā)]m(x1, . . . , xm−1) = e for all x1, . . . , xm−1 that implies rest(f(u), ā) = e. In case ā ∈ An kp the equality [rest(f(u), ā)]m(x1, . . . , xm−1) = = { c ( [u]n+m(a1 mod p, . . . , xm−1 mod p) ) , 06x1, . . . , xm−1<kp, e, otherwise, =      c ( [rest(u, ā mod p)]m(x1 mod p, . . . , xm−1 mod p) ) , 06x1, . . . , xm−1<kp, e, otherwise, = [f(rest(u, ā mod p))]m(x1, . . . , xm−1) holds. Thus rest(f(u), ā) = f(rest(u, ā mod p)). From equality (8) for u ∈ W∞(Gp, Ap) we get Q(u(k)) = Q(f(u)) = {rest(f(u), ā) : ā ∈ An, n > 1} ∪ {f(u)} ⊂ ⊂ {f(rest(u, ā mod p)) : ā ∈ An kp, n > 1} ∪ {e} ∪ {f(u)} ⊂ ⊂ f(Q(u)) ∪ {e} ∪ {f(u)} = f(Q(u)) ∪ {e}. This implies that if u ∈ FW∞(Gp, Ap) then u(k) ∈ FW∞(G,A). Suppose that we have two elements u, v ∈ FW∞(Gp, Ap) that satisfy conditions: 1) u and v are conjugated in the group W∞(Gp, Ap); 2) growth of u is logarithmic; 3) growth of v is exponential. Using elements u and v we construct elements u(k) and v(k). By lemma 4 these new elements belongs to the group FW∞(G,A). Since f(Q(u)) ⊂ Q(f(u)) ⊂ f(Q(u)) ∪ {e}. “adm-n1” — 2019/3/22 — 12:03 — page 68 — #76 68 Conjugacy in finite state wreath powers u and f(u) have equivalent growth. If gu = vg then f(g)f(u) = f(v)f(g). Therefore u and f(u) satisfy the following conditions: 1’) u(k) and v(k) are conjugated in the group W∞(G,A); 2’) growth of u(k) is logarithmic; 3’) growth of v(k) is exponential. Since the growth of an element is invariant under conjugation in FW∞(G,A) (see [2, subsection 4.3]) This implies that elements u(k) and v(k) are non-conjugated in the group FW∞(G,A). Let us consider the following elements of the group FW∞(Gp, Ap) e = [e; e, . . . , e], ai = [σi; a0, a1, . . . , ap−1], 0 6 i < p, s = [σ; e, . . . , e, s], bi = [σi; ai, ai, . . . , ai, bi+1], 0 6 i < p. To simplify notations we will identify bp and b0. Let us show that the elements s and b1 satisfy conditions 1)–3). Lemma 5. An element g ∈ W∞(Gp, Ap) is level transitive (acts transi- tively on the sets Ak p, k > 1) if and only if g∗k = ∏ v∈Ak−1 p [g]k(v) 6= e for all k > 1. Proof. The proof is similar to the proof of lemma 4.4 in [2] and we use two additional facts that the group Gp is abelian and every non-unity element generates a transitive subgroup. Lemma 6. Let p be a odd prime number. Then the element b1 satisfies equalities (b1) ∗ k = σ for all k > 1 which implies that b1 is level transitive. Proof. Equalities (ai) ∗ 1 = (bi) ∗ 1 = σi are obvious and the recurrent formulas (ai) ∗ k+1 = (a0) ∗ k(a1) ∗ k · · · (ap−1) ∗ k, (bi) ∗ k+1 = ((ai) ∗ k) p−1(bi+1) ∗ k follow from definitions. The first of the recurrent formulas implies (ai) ∗ 2 = (a0) ∗ 1(a1) ∗ 1 · · · (ap−1) ∗ 1 = σ0+1+...+(p−1) = σ p(p−1) 2 = e and by induction we get (ai) ∗ k = 1 for all k > 2. The second of the recurrent formulas implies (bi) ∗ 2 = ((ai) ∗ 1) p−1(bi+1) ∗ 1 = σ−iσi+1 = σ, (bi) ∗ k = ((ai) ∗ k−1) p−1(bi+1) ∗ k−1 = (bi+1) ∗ k−1 = σ, k > 3. Lemma 7. The elements s and b1 are conjugated in the group W∞(Gp, Ap). “adm-n1” — 2019/3/22 — 12:03 — page 69 — #77 A. Oliynyk, A. Russyev 69 Proof. The adding machine s is level transitive. The element b1 is level transitive by the lemma 6. Thus the elements s and b1 are conjugated in the group W∞(Sp, Ap). Suppose that equality b1 = g−1sg holds for some g ∈ W∞(Sp, Ap). Let us prove that g ∈ W∞(Gp, Ap). The element g for every k > 0 satisfies equality b pk 1 = g−1sp k g which implies [g]k(ā)(b1) ∗ k = (s)∗k[g]k(ā) for ā ∈ Ak p. From the last equality it follows by lemma 6 that [g]k(ā)σ = σ[g]k(ā) and finally we get [g]k(ā) ∈ Gp. Lemma 8. The element b1 has exponential growth. Proof. The proof is analogous to the proof of the proposition 4.2 in [2]. References [1] P. W. Gawron, V. V. Nekrashevych, V. I. Sushchansky, Conjugation in tree au- tomorphism groups, International Journal of Algebra and Computation, 11, N. 5, 2001, pp. 529–547. [2] R. I. Grigorchuk, V. V. Nekrashevych, V. I. Sushchansky Automata, Dynamical Systems, and Groups, Proceedings of the Steklov Institute of Mathematics, 231, 2000, pp.128 –203. [3] A. V. Russev, On conjugacy in groups of finite-state automorphisms of rooted trees, Ukrainian Mathematical Journal, 60, N. 10, 2008, pp. 1581–1591. [4] I. V. Bondarenko, N. V. Bondarenko, S. N. Sidki, F. R. Zapata, On the conjugacy problem for finite-state automorphisms of regular rooted trees (with an appendix by Raphaël M. Jungers). Groups, Geometry, and Dynamics, 7, 2013, pp. 323–355. [5] A. Oliynyk, Finite state wreath powers of transformation semigroups, Semigroup Forum 82, 2011, pp. 423-436. [6] Y. Lavrenyuk, The group of all finite-state automorphisms of a regular rooted tree has a minimal generating set, Geometriae Dedicata, V.183, Issue 1, 2016, pp. 59–67. Contact information A. Oliynyk Taras Shevchenko National University of Kyiv, Volodymyrska 60, Kyiv, Ukraine, 01033 E-Mail(s): olijnyk@univ.kiev.ua A. Russyev Department of Mathematics, National University of Kyiv-Mohyla Academy, Skovorody St. 2, Kyiv, Ukraine, 04070 E-Mail(s): andrey.russev@gmail.com Received by the editors: 29.01.2019 and in final form 28.02.2019.

Conjugacy in finite state wreath powers of finite permutation groups

Репозитарії

Схожі ресурси