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1
Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 4, P. 135-142
УДК 512.54
DOI 10.46698/l0184-0874-2706-y
ON NORMAL SUBGROUPS OF THE GROUP REPRESENTATION OF THE CAYLEY TREE#
F. H. Haydarov12
1 V. I. Romanovskiy Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan; 2 National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Tashkent 100174, Uzbekistan E-mail: haydarov_imc@mail.ru
Abstract. Gibbs measure plays an important role in statistical mechanics. On a Cayley tree, for describing periodic Gibbs measures for models in statistical mechanics we need subgroups of the group representation of the Cayley tree. A normal subgroup of the group representation of the Cayley tree keeps the invariance property which is a significant tool in finding Gibbs measures. By this occasion, a full description of normal subgroups of the group representation of the Cayley tree is a significant problem in Gibbs measure theory. For instance, in [1, 2] a full description of normal subgroups of indices four, six, eight, and ten for the group representation of a Cayley tree is given. The present paper is a generalization of these papers, i.e., in this paper, for any odd prime number p, we give a characterization of the normal subgroups of indices 2n, n £ {p, 2p} and 2r,i £ N, of the group representation of the Cayley tree.
Keywords: Cayley tree, Gk-group, subgroups of finite index, abelian group, homomorphism. AMS Subject Classification: 20B07, 20E06.
For citation: Haydarov, F. H. On Normal Subgroups of the Group Representation of the Cayley Tree, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 135-142. DOI: 10.46698/l0184-0874-2706-y.
1. Introduction
In group theory, there are some significant open problems, the majority of which arise in solving of problems of sciences such as physics, biology, chemistry, etc. Especially, if the configuration of the particle and lattice system is located on a graph such as lattice, tree, etc (in our case regular tree) then the configuration can be considered as a mapping which is defined on the graph. As usual, the main configurations (mappings) are the periodic ones. It is known that if the graph has a group representation then the periodicity of a mapping can be defined by the given subgroup of the representation. Namely, if H is a given subgroup then we can define a H-periodic mapping, which has a constant value (depending only on the coset) on each (right or left) coset of H. So the periodicity is related to a partition of the group (that presents the graph on which our physical system is located). There are many research manuscripts devoted to several kinds of partitions of groups (lattices) (detail in [3-6]).
# The work supported by the fundamental project (no. F-FA-2021-425) of The Ministry of Innovative Development of the Republic of Uzbekistan.
© 2023 Haydarov, F. H.
The Gibbs measure is a probability measure, which has been an important object in many problems of probability theory and statistical mechanics. In turn, there are many papers which is devoted to periodic and weakly periodic Gibbs measures. In Ref. [5] a bijection between the set of vertices V of the Cayley tree rk and the group Gk is given. Also, a full description of normal subgroups of index two is found and some normal subgroups of the group Gk are constructed. To define periodic and weakly periodic Gibbs measures we need subgroups of Gk. In [7], invariance property of subgroups of group representation of Cayley trees is given and by using this property, the description of the set of periodic or weakly periodic Gibbs measures for Hamiltonians with finite spin values on Cayley trees is reduced to solve the system of algebraic equations. In [8, 9], the problem of describing periodic or weakly periodic Gibbs measures for statistical models on Cayley trees is reduced to solve the system of algebraic equations. If the invariance property holds for any subgroup of the group Gk then we have the opportunity of finding periodic and weakly periodic Gibbs measures corresponding to an arbitrary subgroup of finite index for the group Gk. Also, for any normal subgroup of finite index for the group representation of Cayley tree, the invariance property holds but to study periodic and weakly periodic Gibbs measure we need the exact view of normal subgroups. That is why, we need the description of normal subgroups of finite index (without index two) and to the best of our knowledge there was are full description of a (not normal) subgroup of index 4 of the group representation of the Cayley tree is given in [10]. In [1] and [2] full descriptions of normal subgroups of indices 2i, i € {2,3,4, 5}, for the group Gk are given.
In this paper, we continue this investigation and construct all normal subgroups of index 2n, n € {p, 2p}, and 2^ i € N, for the group representation of the Cayley tree, where p is an odd prime number.
2. Preliminaries
A Cayley tree (Bethe lattice) rk of degree (order) k ^ 1 is a k + 1-regular tree, i.e., a connected, non-directed, acyclic graph with degree of every vertex is k + 1. We denote Cayley tree of degree k + 1 by rk = (V, L) where V and L are the set of vertices and edges respectively.
Let (Gk := (ai, a2,..., ak+1), *) be a group such that o(aj) = 2, i € Nk := {1,2,..., k + 1}, the operation * is a free product. It is known that there exists a bijection from the set of vertices V of the Cayley tree rk to the set of element of the group Gk. To give this correspondence we fix an element x0 € V and let it correspond to the identity element e (i.e., the length of element equals zero) of the group Gk. In positive direction, we label the nearest-neighbors of element e by a1,..., ak+1. Let us label the neighbors of each a^ i = 1,..., k + 1 by a^j, j = 1,..., k + 1. Since all ai have the common neighbor e we have a^ = a2 = e. Other neighbors are labeled starting from aiai in positive direction. We label the set of all the nearest-neighbors of each aiaj- by words aiaj-aq, q = 1,..., k + 1, starting from aiaj-aj = ai by the positive direction. We continue the process and give bijection from the set of vertices V of the Cayley tree rk to the group Gk.
Any (minimal represented) element x € Gk has the following form: x — aii ai2... ain, where 1 ^ im ^ k + 1, m = 1,..., n. The number n is called the length of the word x and is denoted by l(x). The number of letters ai, i = 1,..., k + 1, that enter the non-contractible representation of the word x is denoted by wx(ai).
The following result is well-known in group theory.
Let f be a homomorphism of a group G onto a group G1, HdG such that H C Ker f, and g be the natural homomorphism of G onto G/H. Then there exists a unique homomorphism h of G/H onto G1 such that f = h o g. Furthermore, h is one-one if and only if H = Ker f.
Put ^G and g from G to G/H by g(a) = aH for all a € G. From group theory (e.g., [11]), g is an epimorphism from G onto G/H with Kerg = H.
One of our aim in this paper, we shall give a full description of normal subgroups of finite index of the group Gk.
Let A1,A2,...,Am be subsets Nk and Ai = Aj, for i = j. The intersection is said "contractible" if there exists i0 (1 ^ io ^ m) such that
m f io — 1 \ / m \
O= fl*H n A< •
i=1 \ i=1 / \i=io+1 /
Denote
Ha = < x € Gk : ^ wx(ai) is even > , A c Nk. (1)
I ieA J
We recall main results in [5].
Let A C Nk be a non empty set then Ha < Gk and |Gk : Ha| = 2. Also, for A1, A2,..., Am C Nk if f|i=1 HAi is non-contractible, then f|i=1 HA^ Gk and |Gk ^m=1 HAi I = 2m. One of the important theorem in the book: If H is a subgroup Gk with odd index (= 1) then H is not normal subgroup.
3. Normal Subgroups of Finite Index 2n, n € {p, 2p}, and 21, i € N.
Definition 1 (e.g. [12]). An elementary abelian group (or elementary abelian p-group) is an abelian group in which every nontrivial element has order p. The number p must be prime, and the elementary abelian groups are a particular kind of p-group. The case where p = 2, i.e., an elementary abelian 2-group, is sometimes called a Boolean group.
We denote Boolean group of order 2n by K2n. From group theory it's known that if ^ be a homomorphism of the group Gk onto a finite commutative group G. Then ^>(Gk) is isomorphic to K2i for some i € N.
Indeed, let (G, *) be a commutative group of order n and ^ : Gk G be an epimorphism. Then the group Gk/ Ker ^ has (up to isomorphism) generators and relations (b1,... ,bn : b2 = e1, [bi,bj] = e\), where e1 is an identity element of Gk/Ker^ and [bi,bj] are commutators. This is an elementary abelian group of order 2k. So any homomorphism of Gk into an abelian group is isomorphic to a subgroup of an elementary abelian 2-group, and this is necessarily another elementary abelian 2-group.
Let A1, A2,..., Am c Nk, m € N and P|™1 HAi is non-contractible. Then we denote by Re the following set
Re = i f] HAi : Ai, A2,..., Am c Nk, m € N
The next theorem gives us a family of all subgroups of index 2* of the group Gk coincides with the set Re. For any subgroup H € Re, periodic and weakly periodic Gibbs measures corresponding to H are well studied in [5]. We are going to show that there is not any normal subgroup H of index 2* of Gk such that H € Re.
Theorem 1 [1]. Let ^ be a homomorphism from Gk to a finite commutative group. Then there exists an element H of Re such that Ker ^ ~ H and conversely. Note that, by Theorem 1 we can get easily the following results: Corollary 1 [1, 2]. Any normal subgroup of index 4 has the form Ha H Hb, i. e.
{H : |Gk : H| =4} = {Ha H HB : A, B C Nfc, A = B}.
Any normal subgroup of index 8 has the form Ha H Hb H Hc, i. e.
{H : |Gk : H| =8} = {Ha H HB H Hc : A, B, C C Nk, A = B, B = C, A = C}.
Let G = (b1,b2,...,6r) be a group with free product. If 2n = {A1, A2,..., An} be a partition of Nk\Ao, 0 ^ |Ao| ^ k + 1 — n. Then we define the following homomorphism Un : {a1, a2,..., ak+1} ^ {e1,61..., 6m} given by
Un(x) i e1, if x = ai,i € Ao; (2)
1 6j, if x = ai, i € Aj, j € {1, 2,... , n},
where e1 is the identity element of G.
Put Rb[61,62,..., 6m] is a minimal representation of the word 6. Then we introduce another mapping Yn : G ^ G by the following formula:
{e1, if x = e1;
6i, if x = 6i, i €{1, 2,... , r}; (3)
[61,... ,6r], if x = 6i, i € {1,... ,r}.
Denote
h|,)(G) = {x € Gk : 1(Yn(Un(x))):2p}, 2 < n < k — 1. (4)
We define the following relation on Gk : x ~ y if x = y, where Yn(un(x))) = x. Note that defined relation is an equivalence relation.
Proposition 1. Let be a family of groups of order n which has 2 generators with order two. Then the following equality holds
{Ker : : Gk ^ G € 92n is an epimorphism} Hfj0)BlB2(G) : B1, B2 is a partition of the set Nk\ Bo, 0 < |Bo| < k — 1
< Let G € 92n. We construct a bijection between the two given sets. Note that e1 is the unit element of the group G. For a set B0 c Nk, 0 ^ |B0| ^ k — 1 we take B1,B2 which is a partition of Nk \ B0. Consequently, we can give the homomorphism ^>BoBlB2 : Gk ^ G by the formula
^BoBiB2 (ai) = < f1, if i € B1; (5)
[02, if i € B2.
It's easy to see that for the given subsets B0, B1, B2 we can construct a unique such homomorphism. Also, we have x € Ker ^>BoBlB2 iff x equals e1. Therefore, it is sufficient to prove the following claim: if y € hB0)BiB2 (G) then y = e1. Suppose that there exist y € Gk such that l(y) ^ 2n. Put
y = 6ii6i2 ...6i„, q ^ 2n, S = {6ii,6ii6i2,...,6ii6i2 ...bi„}.
Since S C G there exist x^x2 € S such that x1 = x2 which contradicts the fact that y is a non-contractible. Hence, we showed the inequality l(y) < 2n. From y € H^B^(G) the integer number l(y) have to be divided by 2n. Consequently, we have y = e1 for any
y € Hbbb2(G). For the group G we have Ker (B0B1B2 = hBObb2(G). > Denote
«n := {H(n0)BlB2 (G) : Bi,B2 is apartition of the set \ Bo, 0 < |Bo| < k - 1, |G| = 2nj
u{ Hb)b1b2b3 (G) : Bi, B2, B3 is a partition of the set \ Bo, 0 < |Bo | < k - 2, |G| = 2n
Theorem 2. Let p be an odd prime number. Any normal subgroup of index 2n, n € {p, 2p}, has the form Hg^ (G) U H^B^(G), |G| = 2n i. e.,
«n = {H : ff<Gfc, |Gfc : H| = 2n}.
< At first we prove that
«n C{H : H<Gfc, |Gk : H| = 2n}.
Let G be a finite group and the number of elements is 2n. Also, B1,B2 is a partition of \ Bo, 0 < |Bo| < k - 1. It is enough to show that x-1HBn0)BlB2(G) x C HBn0BlB2(G), for
all x € Gk. Similar to the proof of Proposition 1, we can conclude that if y € H^bb (G) then y = e1, where e1 is the identity element of G. If we take an element z from the coset x-1H(n)BlB2 (G) x, then z = x-1h x for some h € H(n)BlB2 (G). Consequently, one gets
Z = Yn(vn(z)) = Yn (vn(x-1h x)) = Yn (vn(x-1)vn(h)vn(x))
= Yn (vn(x-1)) Yn (Vn(h)) Yn (Vn(x)) = (Yn (Vn(x)))-1 Yn (Vn(h)) Yn (vn(x)).
Since Yn (vn(h)) = e1 we have i = e1, i.e., z € hB0BiB2(G). Namely
H(n)BiB2(G) € {H : H < Gfc, |Gfc : H| = 2n} .
Now we show that {H : H < Gfc, |Gfc : H| = 2n} C «n. Put H < Gfc, |Gfc : H| = 2n. We consider a natural homomorphism 0 : Gk ^ Gk : H, i.e., 0(x) = xH, x € Gk. We can find elements: e, b1, b2,..., b2n-1 such that 0 : Gk ^ {H, b1 H,..., b2n-1 H} is an epimorphism. Let ({H, b1 H,..., b2n-1 H}, *) = p, i. e., p is the factor group. If we show that p € 92n then the theorem will be proved. Assuming that p € 92n, then there are at least three generators: c1, c2,..., cq € p, q ^ 3, such that p = (c1,..., cq}. Clearly, (c1, c2) is a subgroup of p and elements of the group (c1, c2)| are greater than 3. By Lagrange's theorem and n € {p, 2p}, we obtain |(c1 ,c2}| €{4, 2p, 4p} .
Let us consider the case: |(c1,c2}| = 4. If the number four isn't equal to one of these numbers |(c1,c3)| or |(c2,c3}| then we shall take these pairs. If |(c1,c2)| = |(c1 ,c3}| = |(c2,c3}| = 4, then elements of the group (c1 ,c2,c3} is 8. We use Lagrange's theorem and conclude |p| = 2n is divided by eight, i.e., it is impossible. For the case n = p, since Lagrange's theorem one gets:
|(ci,c2)| € \ m : — € N m
If e2 is the identity element of p, then from c2 = e2 we take |(c1, c2)| = 2n. Consequently, we have (c1,c2) = p, but the second handside, c3 / (c1,c2). Hence, p € 92n.
Finally, we consider the case n = 2p. Again by Lagrange's theorem we obtain
|<C1,C2)| e \m : — € N m
Let e2 be the unit element of p. Then since c2 = e2 one gets |(c1, c2)| = 2n. Consequently, (c1,c2) = p which contradicts to c3 / (c1,c2). Hence, p € 92n. If |(c1,C2)| = 2p, then
(ci, c2) = {e, ci, c2, cic2, cic2ci,..., cic2^.. cij = A.
2(P-I)
It's easy to check that
c3A U A C p, c3A H A = 0, |c3A U A| = |c3A| + |A| = 2n = |p|.
We then deduce that c3A U A = p. On the second hand side, we showed that c3c1c3 € p does not belong to c3A U A. Clearly, from c1,c2,c3 are generators, our conclusion is c3c1c3 / A. Thus, c3c1c3 € c3A ^ c3c1c3 = c3x with x € (c1,c2). But x = c1c3 / (c1,c2). If | (C1, C2)| = 4p, then (c1, C2) = p, but C3 / (c1,c2). Hence p € >
As a corollary of Theorem 2, we give the main theorems in [1, 2], i.e., let Sn = {A1, A2,..., An} be a partition of {1,2,..., k + 1}\A0, 0 ^ |A0| ^ k + 1 — n, and it is considered function u« : {a1, a2,..., ak+1} ^ {e, a1..., ak+1} as
. I e, if x = ai, i € A0; un(x) = <
I amj, if x = ai, i € Aj, j = 1,2,... ,n. Define Yn : Gk ^ Gk by the formula
Yn(x) = Yn(aiiai2 ... ais) = u«(aii)u«(ai2)... u„(ais).
Put
H^i) = {x : 1(Yn(x)) : 2i}, n<k + 1, i € {3, 5}.
Note that these corollary is not so difficult in group theory but our main aim is to feel elements of these subgroups as vertices of Cayley tree. Only in this case we have a chance to study periodic and weakly periodic Gibbs measures on Cayley trees.
Corollary 2 [1, 2]. Let H be a normal subgroup of the group Gk. Then
1. {H(3) : |Gk : H| =6} = {H^, H^};
2. {H(4) : |Gk : H| =8} = {HS2, HS3}.
Remark. In general, we can not say any normal subgroup of index 2i, i € N, has the form Hii), n € N.
References
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Received June 23, 2022
Farhod H. Haydarov V. I. Romanovskiy Institute of Mathematics, 9 University St., Tashkent 100174, Uzbekistan, Post Doctoral Students;
National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Tashkent 100174, Uzbekistan, Docent, Department of Algebra and Functional Analysis E-mail: haydarov_imc@mail.ru https://orcid.org/0000-0001-9388-122X
Владикавказский математический журнал 2023, Том 25, Выпуск 4, С. 135-142
О НОРМАЛЬНЫХ ПОДГРУППАХ ГРУППОВОГО ПРЕДСТАВЛЕНИЯ ДЕРЕВА КЭЛИ
Хайдаров Ф. Х.1,2
1 Институт математики имени В. И. Романовского, Узбекистан, 100174, Ташкент, ул. Университетская, 9;
2 Национальный университет Республики Узбекистан им. М. Улугбека, Узбекистан, 100174, Ташкент, ул. Университетская, 4 E-mail: haydarov_imc@mail.ru
Аннотация. Мера Гиббса играет важную роль в статистической механике. На дереве Кэли для описания периодических мер Гиббса для моделей статистической механики нам нужны подгруппы группового представления дерева Кэли. Нормальная подгруппа группового представления дерева Кэли сохраняет свойство инвариантности, которое является важным инструментом при поиске мер Гиббса. В связи с этим полное описание нормальных подгрупп группового представления дерева Кэли является важной проблемой теории меры Гиббса. Например, в [1, 2] дано полное описание нормальных подгрупп индексов четыре, шесть, восемь и десять для группового представления дерева Кэли. Настоящая работа является обобщением этих работ, т. е. в ней для любого нечетного простого числа p дается характеризация нормальных подгрупп индексов 2n, n £ {p, 2p} и 2г,i £ N, группового представления дерева Кэли.
Ключевые слова: дерево Кэли, Gk-группа, подгруппы конечного индекса, абелева группа, гомоморфизм.
AMS Subject Classification: 20B07, 20E06.
Образец цитирования: Haydarov F. H. On Normal Subgroups of the Group Representation of the Cayley Tree // Владикавк. матем. журн.—2023.—Т. 25, № 4.—C. 135-142 (in English). DOI: 10.46698/10184-0874-2706-y.
