On normal subgroups of the group representation of the Cayley tree

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.

2.    Preliminaries

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 Г ^к and the group G k is given. Also, a full description of normal subgroups of index two is found and some normal subgroups of the group G k are constructed. To define periodic and weakly periodic Gibbs measures we need subgroups of G _k . 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 G k then we have the opportunity of finding periodic and weakly periodic Gibbs measures corresponding to an arbitrary subgroup of finite index for the group G k . 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 E { 2, 3,4, 5 } , for the group G k are given.

In this paper, we continue this investigation and construct all normal subgroups of index 2n , n E { p, 2p } , and 2 i , i E N, for the group representation of the Cayley tree, where p is an odd prime number.

A Cayley tree (Bethe lattice) Г ^к 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 Г ^к = (V, L) where V and L are the set of vertices and edges respectively.

Let ( G k := ( a i , a 2 , ..., a k +i ) , * ) be a group such that o ( a i ) =2 , i E N k := { 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 r ^k to the set of element of the group G k . To give this correspondence we fix an element x o E V and let it correspond to the identity element e (i. e., the length of element equals zero) of the group G _k . In positive direction, we label the nearestneighbors of element e by a i ,..., a k +i . Let us label the neighbors of each a i , i = 1,..., k + 1 by a i a j , j = 1,..., k + 1 . Since all a i have the common neighbor e we have a i a i = a ² = e. Other neighbors are labeled starting from a i a i in positive direction. We label the set of all the nearest-neighbors of each a i a j by words a i a j a q , q = 1,..., k +1, starting from a i a j a j = a i by the positive direction. We continue the process and give bijection from the set of vertices V of the Cayley tree r ^k to the group G k .

Any (minimal represented) element x E G k has the following form: x = a i 1 a i ₂ ...a i n , where 1 ^ i m ^ 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 a i , i = 1,..., k + 1, that enter the non-contractible representation of the word x is denoted by w _x (a i ).

The following result is well-known in group theory.

Let f be a homomorphism of a group G onto a group G i , H<\G 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 G i such that f = h о g . Furthermore, h is one-one if and only if H = Ker f .

Put H < G and g from G to G/H by g(a) = aH for all a E G . From group theory (e. g., [11]), g is an epimorphism from G onto G/H with Ker g = H .

One of our aim in this paper, we shall give a full description of normal subgroups of finite index of the group G _k .

Let A i ,A 2 ,...,A _m be subsets N k and A i = A j , for i = j. The intersection is said “contractible” if there exists i g (1 ^ i g ^ m) such that

m n Ai = i=i

( i o — i \     / m

A i .

n A i n П i =i   /     \i = i o +1

Denote

H a = x E G k : ^2^_x(a i ) is even , i ∈ A

A C N k .

We recall main results in [5].

Let A C N k be a non empty set then H a < G k and | G k : H a | = 2 . Also, for A i ,A 2 ,...,A _m C N k if Plm ^i H A i is non-contractible, then PlmP H A i <1 G k and | G k : Pm i H A i | = 2 m . One of the important theorem in the book: If H is a subgroup G k with odd index (= 1) then H is not normal subgroup.

3.    Normal Subgroups of Finite Index 2n, n E {p, 2p}, and 2i, i E 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 2 ⁿ by K 2 n . From group theory it’s known that if ^ be a homomorphism of the group G k onto a finite commutative group G. Then ^ ( G k ) is isomorphic to K ₂ i for some i E N .

Indeed, let (G, * ) be a commutative group of order n and ^ : G k ^ G be an epimorphism. Then the group G k / Ker ф has (up to isomorphism) generators and relations b i ,..., b n : b ² = e i , [b i ,b j ] = e i ), where e i is an identity element of G k / Ker ^ and [b i , b j ] are commutators. This is an elementary abelian group of order 2 ^k . So any homomorphism of G _k 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 A i , A 2 ,..., A m C N k , m E N and Pmi H A i is non-contractible. Then we denote by Re the following set

Re =

m n i =i

H A i : A i ,A 2 ,.

A „

m

C N k , m E N

The next theorem gives us a family of all subgroups of index 2 ⁱ of the group G _k coincides with the set Re . For any subgroup H E 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 ⁱ of G k such that H E / Re .

Theorem 1 [1] . Let ϕ be a homomorphism from G k 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 П Hb, i. e.

{ H : | G k : H | = 4 } = { H a П H b : A, B C N k , A = B } .

Any normal subgroup of index 8 has the form Ha П Hb П He, i. e.

{ H : | G k : H | = 8 } = { H a П H b П H e : A,B,C C N k , A = B, B = C, A = C } .

Let G = ( b i , b 2 ,..., b _r ) be a group with free product. If H n = { A i , A 2 ,..., A n } be a partition of N k \ A o , 0 C | A o | C k + 1 - n. Then we define the following homomorphism U n : { a i , a 2 ,..., a k +i } ^ { e i , b i ..., b m } given by

U n (x) =

I

if x = ai, i E A o ;

if x = a i , i E A j , j E { 1, 2,... , n } ,

where e 1 is the identity element of G .

Put R b [b i , b 2 ,..., b m ] is a minimal representation of the word b . Then we introduce another mapping Y n : G ^ G by the following formula:

e 1 ,

Y n ( x ) =    b i ,

if if

R b i [b i ,...,b r ], if

x = e _i ;

x = b i , i E { 1, 2,... ,r } ;

x = b i , i E { 1,... ,r } .

Denote

H = ^P _n ) (G) = { x E G k : l(Y n (u n (x))) : 2p } , 2 C n C k - 1.                 (4)

We define the following relation on G k : x ~ y if x = y, where Y n (u n (x))) = x. Note that defined relation is an equivalence relation.

Proposition 1. Let ^n be a family of groups of order n which has 2 generators with order two. Then the following equality holds

{ Ker ^ : ^ : G k ^ G E ^ _{2 n} is an epimorphism }

= {H Bn ) B 1 B ₂ (G) : B _i ,B ₂ is a partition of the set N k \ B o , 0 C | B o | C k - 1} •

• <1 Let G E ^ 2 n . We construct a bijection between the two given sets. Note that e i is the unit element of the group G . For a set B o C N k , 0 C | B o | C k - 1 we take B i ,B 2 which is a partition of N k \ B o . Consequently, we can give the homomorphism ^ в _о в 1 в ₂ : G k ^ G by the formula

  V B 0 B 1 B 2 ^(a i ) = I

  
  

b i , if i E B i; b 2 , if i E B 2 .

It’s easy to see that for the given subsets B0 , B1, B2 we can construct a unique such homomorphism. Also, we have x E Ker^вов1в2 iff x equals ei. Therefore, it is sufficient to prove the following claim: if y E ^(^1^2 (G) then yj = ei. Suppose that there exist y E Gk such that l(y) ^ 2n. Put y = bii bi^ ...biq, q ^ 2n, S = {bii ,bii bi2 ,•••,bi1 bi2 ...biq}.

Since S C G there exist x i ,x 2 € S such that x i = X 2 which contradicts the fact that y is a non-contractible. Hence, we showed the inequality l(y) < 2n. From y € H^ ⁽ ngB ₁ B ₂ (G) the integer number l(y) have to be divided by 2n. Consequently, we have y = e i for any У ^E HR^ b ₂ ^{( G )} . ^{For the} grouP ^{G we have Ker} ^ B q B i B 2 = ^H b 0 , _{B 1} b ₂ ^{( G )}- ^>

Denote

^ „ := { н В”в 1 в ₂ (G) : B _i ,B ₂ is a partition of the set N \ B q , 0 C | B q | C k — 1, | G | = 2n |

U |H B 0 ) B 1 B ₂ B ₃ (G) : B _i , B ₂ , B 3 is apartition of the set N \ B q , 0 C | B q | C k — 2, | G | = 2n| .

Theorem 2. Let p be an odd prime number. Any normal subgroup of index 2n, n 6 {p, 2P}, has the form HR^R^ (G) U HBnoB1B2B3 (G), |G| = 2n i. e.,

K n = { H : Hk, |Gk : H| = 2n}.

<1 At first we prove that

^nC{ H : H <] G k , | G k : H | = 2n } .

Let G be a finite group and the number of elements is 2n. Also, Bi , B2 is a partition of Nk \ Bq, 0 C |Bq| C k — 1. It is enough to show that x^H^^ b2 (G) x C H^b-^ (G), for all x € Gk. Similar to the proof of Proposition 1, we can conclude that if y € H^EngB1B2 (G) then y = ei, where ei is the identity element of G. If we take an element z from the coset x-1HbOb1B2 (G) x, then z = x ih x for some h € H^BjgB1B2 (G). Consequently, one gets z = Yn(vn(z)) = Yn Vnnxc^h x)) = Yn kpx ' wph UV(x))

= Y n (v n (x ^-1 )) Y n (v n (h)) Y n (v n (x)) = (Y n (v n (x))) ^-i Y n (v n (h)) Y n (v n (x)) .

Since Y n (v n (h)) = e _i we have z = e _i , i. e., z € H Bn ₀)_B1 B ₂ ( G ) . Namely

H BnB i B 2 (G) €{ H : H < G k , G : H | = 2n } .

Now we show that { H : H < G k , | G k : H | = 2n } C K _n . Put H < G k , | G k : H | = 2n. We consider a natural homomorphism ф : G k ^ G k : H , i.e., ф(x) = xH , x € G k . We can find elements: e, b i , b 2 ,..., b 2 n - i such that ф : G k ^ { H, b i H,..., b 2_{n -} i H } is an epimorphism. Let ( { H, b i H,..., b 2 _{n -} i H } , * ) = p, i. e., p is the factor group. If we show that p € ^ 2 n then the theorem will be proved. Assuming that p / ^ 2 n , then there are at least three generators: c i , C 2 ,..., c _q € p , q ^ 3 , such that p = ( c i ,... , c _q ) . Clearly, ( c i , C 2 ) is a subgroup of p and elements of the group ( c i , C 2 )| are greater than 3. By Lagrange’s theorem and n € { p, 2p } , we obtain |( c i , c 2 )| € { 4, 2p, 4p } .

Let us consider the case: |( c i ,c 2 )| = 4. If the number four isn’t equal to one of these numbers |( c i ,c 3 )| or |( c 2 ,c 3 )| then we shall take these pairs. If |( c i ,c 2 )| = |( c i , c 3 )| = |( c 2 ,c 3 )| = 4 , then elements of the group ( c i , c 2 ,c 3 ) 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:

^|( c i , c 2 >| ^€ I

m :

2p € n) . m

If e 2 is the identity element of p , then from c i = e 2 we take | ( c i , C 2 >| = 2n. Consequently, we have ( c i ,c ₂ ) = p , but the second handside, C 3 € ( c i ,c ₂ > . Hence, p € ^ 2 n .

Finally, we consider the case n = 2p . Again by Lagrange’s theorem we obtain

|( c i ,c 2 >| € Im : — € n) . m

Let e 2 be the unit element of p . Then since c i = e 2 one gets | ( c i , C 2 > | = 2n. Consequently, ( c i ,C 2 > = P which contradicts to c 3 / ( c i ,c ₂ ) . Hence, p E ^ 2 n .

If | ( c i , c 2 > | = 2p , then

( c i ,c 2 > = { e,c i ,c 2 ,c i c 2 ,c i c 2 c i , . • • , c i c2^ • c i } = A.

2( p - 1)

It’s easy to check that c3A U A c p, c3A n A = 0, ^A U A| = |caA| + |A| = 2n = |p|.

We then deduce that c 3 A U A = p . On the second hand side, we showed that c 3 c i c 3 € p does not belong to c 3 A U A. Clearly, from c i ,c ₂ ,c 3 are generators, our conclusion is c 3 c i c 3 / A . Thus, c 3 c _i c 3 € c 3 A ^ c 3 c _i c 3 = c 3 x with x € ( c _i ,c ₂ > . But x = c _i c 3 / ( c _i ,c ₂ > .

If |( c i , c 2 >| = 4p , then ( c i , c 2 > = p , but c 3 / ( c i ,c 2 > . Hence p E ^n >

As a corollary of Theorem 2, we give the main theorems in [1, 2], i.e., let S _n = { A i , A ₂ ,..., A n } be a partition of { 1, 2,..., k + 1 }\ A q , 0 C | A o | C k + 1 — n, and it is considered function u _n : { a i , a _{2 )}..., a k +i } ^ { e, a i ..., a k +i } as

U n (x) =

I

if x = a i , i € A o ;

if x = a i , i € A j , j = 1, 2,.

n.

Define Y n : G k ^ G k by the formula

^Y n ^{( x)} — ^Y n ^{( a} i i ^a i 2 . . . a i s ) — u n ^(a i i ^{) u} n ^{( a} i 2 ) ... u n ^(a i s ^{) .}

Put

H i ) = { x : l(Y n (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 G k . Then

• 1.    { H ⁽³⁾ :   | G k : H | = 6 } = { H  , H = 3 } ;

• 2.    { H ⁽⁴⁾ :   | G k : H | = 8 } = { H  , H = 3 } .

Remark. In general, we can not say any normal subgroup of index 2i , i € N , has the form H ^{( i )} , n € N.

Ξ _n