Automorphisms of the category of finitely generated free groups of the certain subvariety of the variety of all groups

Programa de Pós-Graduação em Matemática Aplicada e Estatística

Mestrado em Matemática Aplicada e Estatística

Automorphisms of the category of nitely

generated free groups of the certain subvariety

of the variety of all groups

Ruan Barbosa Fernandes

Natal-RN March 2020

Automorphisms of the category of nitely generated

free groups of the certain subvariety of the variety of

all groups

Trabalho apresentado ao Programa de Pós-Graduação em Matemática Aplicada e Es-tatística da Universidade Federal do Rio Grande do Norte, em cumprimento com as exigências legais para obtenção do título de Mestre.

Área de Concentração: Modelagem Matemá-tica.

Linha de Pesquisa: Matemática Computaci-onal

Orientador

Professor Dr. Arkady Tsurkov

Co-orientadora

Professora Dra. Elena Aladova

Universidade Federal do Rio Grande do Norte  UFRN

Programa de Pós-Graduação em Matemática Aplicada e Estatística  PPGMAE

Natal-RN March 2020

free groups of the certain subvariety of the variety of all groups apresentada por Ruan Barbosa Fernandes e aceita pelo Programa de Pós-Graduação em Matemática Aplicada e Estatística da Universidade Federal do Rio Grande do Norte, sendo aprovada por todos os membros da banca examinadora abaixo especicada:

Professor Doutor Arkady Tsurkov

Orientador

Departamento de Matemática

Universidade Federal do Rio Grande do Norte - UFRN

Professora Doutora Elena Aladova

Co-orientadora

Departamento de Matemática

Universidade Federal do Rio Grande do Norte - UFRN

Professor Doutor Alexey Kuzmin

Departamento de Matemática

Universidade Federal do Rio Grande do Norte - UFRN

Professor Doutor Eugene Plotkin

Departamento de Matemática Universidade Bar-Ilan, Israel.

Unknowlegments

We acknowledge Prof. A. I. Reznokov from St.-Petrsburg State University, which provide to authors the copy of (SANOV, 1940).

We are thankful to Professor emeritus of UFRN N. Cohen for his important remarks, which helped a lot in writing this thesis.

is like mining radium. For every gram you work a year. For the sake of a single word you waste a thousand tons of verbal ore. V. V. Mayakovsky

nitamente gerados de uma subvariedade da variedade

de todos os grupos

Autor: Ruan Barbosa Fernandes Orientador: Professor Doutor Arkady Tsurkov Co-orientadora: Professora Doutora Elena Aladova

Resumo

Em geometria algébrica universal, a categoria Θ0 _{das álgebras livres nitamente geradas}

de alguma variedade fíxa Θ de álgebras e o grupo quociente A/Y são muito importantes. Aqui A é o grupo de todos os automorsmos da categoria Θ0 e Y é o grupo de todos o

automorsmos internos de Θ0_{. Na variedade de todos os grupos, todos os grupos abelianos}

(PLOTKIN; ZHITOMIRSKI, 2006), todos os grupos nilpotentes de classe n (n ≥ 2) ( TSUR-KOV, 2007b) o grupo A/Y é trivial. B. Plotkin propôs a seguinte pergunta: Existe uma

subvariedade da variedade de todos os grupos tal que o grupo A/Y nessa subvariedade não seja trivial? A. Tsurkov supôs que existe alguma variedade de grupos periódicos, tal que o grupo A/Y nessa variedade não é trivial. Neste trabalho, nós damos um exemplo de uma subvariedade deste tipo.

Palavras-chave: Geometria algébrica universal, teoria de categoria, equivalência automór-ca, grupos nilpotentes, grupos periódicos.

free groups of the certain subvariety of the variety of

all groups

Author: Ruan Barbosa Fernandes Advisor: Professor Doctor Arkady Tsurkov Co-advisor: Professor Doctor Elena Aladova

Abstract

In universal algebraic geometry, the category Θ0_{of nitely generated free algebras of some}

xed variety Θ of algebras and the quotient group A/Y play a central role. Here A is the group of automorphisms of the category Θ0 and Y is the group of inner automorphisms

of this category. In the varieties of all groups, all abelian groups (PLOTKIN; ZHITOMIRSKI,

2006), all nilpotent groups of the class no more then n (n ≥ 2) (TSURKOV, 2007b) the

group A/Y is trivial. B. Plotkin posed the question whether there exists a subvariety of the variety of all groups, such that the group A/Y in this subvariety is not trivial. A. Tsurkov hypothesized that for some varieties of periodic groups, the groups A/Y is not trivial. In this work we give an example of one particular subvariety of this kind.

Keywords: Universal algebraic geometry, category theory, automorphic equivalence, nil-potent groups, periodic groups.

Sumário

1 Introduction p. 11 1.1 Motivation . . . p. 11 1.2 Thesis Overview . . . p. 13 2 Preliminaries p. 16 2.1 Universal Algebra . . . p. 16 2.2 Variety of algebras . . . p. 22 2.3 Category . . . p. 25 3 The method of verbal operations p. 28 3.1 First denitions and basic facts . . . p. 28 3.2 Strongly stable automorphism and strongly stable system of bijections . p. 29 3.3 Strongly stable system of bijections and applicable systems of words . . p. 30 3.4 Automorphisms, which are strongly stable and inner . . . p. 32 4 Application of the method of verbal operations. p. 34 5 Some properties of the varieties N4 and Θ p. 36

6 Some lemmas about the group N4(x, y) /R p. 43

7 Computation of the group FΘ(x, y) p. 45

8 Applicable systems of words. Necessary conditions p. 49 9 Applicable systems of words. Sucient conditions p. 57

9.2 System of words W2 . . . p. 59

9.3 System of words W3 . . . p. 61

10 The group S ∩ Y and the group A/Y p. 63

11 Final Considerations p. 65

11.1 An open problem . . . p. 65

1 Introduction

This work is devoted to some aspects of universal algebraic geometry, i.e., geometry over universal algebras (for denition of universal algebra see, for example (KUROSH,

1965)).

All denitions of the basic notions of universal algebraic geometry can be found, for example, in (PLOTKIN, 1997), (PLOTKIN, 1998), (PLOTKIN, 2003) and (PLOTKIN; PLOT-KIN, 2015). Additional fundamental papers are (BAUMSLAG; MYASNIKOV; REMESLEN-NIKOV, 1999), (MYASNIKOV; REMESLENNIKOV, 2000) and (DANIYAROVA; MYASNIKOV; REMESLENNIKOV, 2012a), (DANIYAROVA; MYASNIKOV; REMESLENNIKOV, 2012b).

1.1 Motivation

One of the natural question of universal algebraic geometry is the following:

Problem 1.1 When do two algebras H1 and H2 from some variety of algebras Θ have

the same algebraic geometry?

By this we mean an isomorphism of the categories of algebraic closed sets over H1

and H2, respectively. So, Problem 1.1 is ultimately related to the following one:

Problem 1.2 What are the conditions which provide an isomorphism of the categories of algebraic closed sets over the algebras H1 and H2?

Notions of geometric and automorphic equivalences of algebras play here a crucial role. In universal algebraic geometry we consider some variety Θ of universal algebras of signature Ω. We denote by X0 an innite countable set of symbols. By F(X0)we denote the

set of all nite subsets of X0. We will consider the category Θ0, whose objects are all the

of the category Θ0 _{are homomorphisms of such algebras. We will occasionally denote}

F (X) = F (x1, x2, . . . , xn) if X = {x1, x2, . . . , xn}. We consider a system of equations

T ⊆ F × F, where F ∈ ObΘ0, and we solve these equations in an arbitrary algebra

H ∈ Θ. The set Hom(F, H) serves as an ane space over the algebra H: the solution of the system T is a homomorphism µ ∈ Hom(F, H) such that µ(t1) = µ(t2)holds for every

(t1, t2) ∈ T (or T ⊆ ker µ). The set

T_H0 = {µ ∈Hom(F, H)|T ⊂ ker µ}

will be the set of all solutions of the system T . We call these sets algebraic, as in classical algebraic geometry. For every set of points R ⊆ Hom(F, H) we consider a congruence of equations dened in this way:

R0_H = \

µ∈R

ker µ.

This is a maximal system of equations which has R as its set of solutions. For every set of equations T we consider its algebraic closure T00

H =

T

µ∈T_H0 ker µ with respect to the

algebra H. A set T ⊆ F × F is called H-closed if T = T00

H. An H-closed set is always a

congruence. We denote the family of all H-closed congruences in F by ClH(F ).

Denition 1.1 Algebras H1, H2 ∈ Θ are geometrically equivalent if for every F ∈

ObΘ0 _{and every T ⊆ F × F the equality T}00 H1 = T

00

H2 is fullled.

By this denition, algebras H1, H2 ∈ Θare geometrically equivalent if and only if the

families ClH1(F ) and ClH2(F ) coincide for every F ∈ ObΘ

0_.

A concept of geometric similarity of algebras is ner than the previous. The denition of this concept can be ned in (PLOTKIN, 1998) and (PLOTKIN, 1998). All necessary denition from universal algebra and Category theory are presented in Chapter 2.

We consider two algebras H1, H2 ∈ Θ. By Θ1 and Θ2 we denote the varieties generated

by algebras H1 and H2 respectively. We can construct functors ClHi : Θ

0

i → P oset from

the categories Θ0

i to the category P oset of partially ordered sets (i = 1, 2). We say that

an isomorphism Φ : Θ0

1 → Θ02 provide the geometric similarity of algebras H1 and H2 if

the functor ClH1 is isomorphic to the functor ClH2Φ. In this situation in a particular case,

when Θ1 = Θ2 = Θ, we say that an automorphism Φ : Θ0 → Θ0 provide the automorphic

equivalence of algebras H1 and H2.

The denition of automorphic equivalence in the language of the category of coordi-nate algebras was considered in (PLOTKIN, 2003) and (TSURKOV, 2016). Intuitively, the

algebras H1, H2 ∈ Θ are automorphically equivalent if and only if the families ClH1(F )

and ClH2(Φ(F )) coincide up to a change of coordinates. This change is dened by the

automorphism Φ.

Denition 1.2 An automorphism Υ of an arbitrary category K is inner, if it is iso-morphic as a functor to the identity automorphism of the category K.

Thus means that for every F ∈ ObK there exists an isomorphism σΥ

F : F −→ Υ(F ) such

that for every µ ∈ MorK(F1, F2)

Υ(µ) = σΥ_F2µ(σ_FΥ₁)−1

holds. It is clear that the set Y of all inner automorphisms of an arbitrary category K is a normal subgroup of the group A of all automorphisms of this category. If an inner automorphism Υ provides the automorphic equivalence of the algebras H1 and H2,

where H1 ,H2 ∈ Θ, then H1 and H2 are geometrically equivalent (see (PLOTKIN, 2003)).

Therefore the quotient group A/Y measures the gap between geometric equivalence and automorphic equivalence of algebras from the variety Θ: if the group A/Y is trivial, then the notions of geometric equivalence and the automorphic equivalence coincide in the variety Θ. The converse is not true. For example, in the variety of linear spaces over some xed eld k we have that A/Y ∼= Autk, where Autk is the group of all automorphisms of the eld k. The proof of this fact can be obtained by the method of (TSURKOV, 2014).

But all linear spaces over a xed eld k are geometrically equivalent. This fact is a simple conclusion from (PLOTKIN; PLOTKIN; TSURKOV, 1999). In the varieties of groups, abelian

groups (PLOTKIN; ZHITOMIRSKI, 2006), nilpotent groups of class no more then n (n ≥ 2)

(TSURKOV, 2007b) the group A/Y is trivial, so geometric equivalence and automorphic

equivalence coincide in these varieties. B. Plotkin posed the question whether there exists a subvariety of the variety of groups, for which the group A/Y is not trivial. A. Tsurkov hypothesized that these exists some varieties of periodic groups, such that the groups A/Y in these varieties is not trivial. Our main objective is to conrm this hypothesis.

1.2 Thesis Overview

In universal algebraic geometry, considering any variety Θ of algebras, let Θ0 _{be the}

category of nitely generated free algebras in Θ; let A be the group of automorphisms of the category Θ0_{; and let Y be the subgroup of inner automorphisms. A natural question is}

to calculate the quotient group A/Y, and especially determine if it is trivial, i.e. if every automorphism of Θ0 _{is inner, because the quotient group A/Y measures the possible}

dierence geometric and automorphic equivalences of algebras of the variety Θ. Hence, if the quotient group A/Y is trivial, we can a priori conclude that in the variety Θ the geometric and automorphic equivalences of algebras are coincide.

B. Plotkin posed the question:

"Whether the quotient group A/Y can be non-trivial for certain variety of groups?"And Tsurkov hypothesized that the answer may be positive for some varieties of periodic groups. In this thesis we construct one particular variety of this type, namely, the variety Θof groups with the following joint three characteristics: nilpotent of class 4, metabelian, and Sanov. For the precise denition see (2.6)  (2.8).

Our main result is that in the variety Θ the group A/Y is not trivial, in fact, has order 2. This will be done by applying the method of verbal operations elaborated in (PLOTKIN; ZHITOMIRSKI, 2006) and (TSURKOV, 2016). As a by-product of this analysis we obtain a concrete representation for the outer (i.e. non-inner) automorphism in the variety Θ.

In Chapter 2 we introduce the basic denitions and concepts from universal algebra, varieties of algebras, and category theory.

In Chapter 3 we discuss the method of verbal operations and its consequences. A central role is played by a Condition 3.1. By (PLOTKIN; ZHITOMIRSKI, 2006), if an arbitrary variety Θ fullls this condition, then every automorphism of the category Θ0

can be decomposed as the composition of a strongly stable automorphism and an inner automorphism (see Denition 1.3 and Denition 3.2). So, for the computation of the group A/Y it is enough to compute the groups S and S ∩ Y, where S is the group of all strongly automorphisms of the category Θ0.

The method of verbal operations provides a bijection between the set of all stron-gly stable automorphisms of our category Θ0 and the set of all applicable systems of

words (elements of nitely generated free algebras) (see Subchapter 3.3). In addition, this method provides a criterion when a strongly stable automorphism is an inner au-tomorphism (see Criterion 3.1 and Proposition 3.1). Hence this method provides all the theoretical ingredients needed to solve the problem.

We show in Chapter 4 that our particular variety, dened in (2.6), (2.7) and (2.8) fullls the IBN condition, hence, fullls the Condition 3.1. So we can apply the method of verbal operations for computation of the group A/Y of this variety.

From Chapter 5 to the end of the thesis, we denote by Θ our particular variety, dened in (2.6), (2.7) and (2.8). Chapters 5, 6 and 7 are devoted to the computation of the doubly generated free group of our variety Θ: FΘ(x, y).

In Chapter 5, we compute the collection formula which is a result of collection process (see HALL, 1959, 11.1 and 18.2) of two elements for nilpotent class 4 groups (see (5.11)). Thereafter we conclude from this formula and from other identities of our variety Θ some set of relations in FΘ(x, y) (see (5.21) and (5.22)). After this we class 4 group N4(x, y).

We denote by R the minimal normal subgroup generated by these elements and elements x4_{, y}4_.

In Chapter 6 we prove some lemmas about the quotient group N4(x, y)/R. These

lemmas will be used in Chapter 7.

In Chapter 7 we prove that FΘ(x, y) ∼= N4(x, y)/R.

The next two chapters are devoted to the determination of the applicable systems of words for our variety. These systems must contain the elements of FΘ(x, y), so we need

use the results of previous chapters.

In Chapter 8 we consider the necessary conditions for the systems of words to be applicable. We prove that only four dierent systems of words can be applicable. In Chapter 9 we prove that all the four systems are indeed applicable.

In the beginning of Chapter 10 we prove that only two automorphisms dened by these systems of words are inner. With this we conclude that for our variety the order of the group A/Y is equal to two.

2 Preliminaries

In this section we provide preliminary denitions about universal algebras, groups, category and variety.

2.1 Universal Algebra

Denition 2.1 An universal algebra is a pair (A, Ω), where A is a non-empty set and Ω is a collection of operations over A. The collection Ω is called the signature of the algebra A.

Operations are dened as follows

Denition 2.2 Let A be a non-empty set. We dene: 1. A0

= ∅;

2. For any positive integer n, An _{is the carthesian product of n copies of A , that is,}

an element a ∈ An has the form a = (a

1, . . . an), where ai ∈ A, 1 ≤ i ≤ n.

3. A mapping ω : An _{→ A} is called n-ary operation over A. The number n is called

arity (or rank) of ω, denote by ρω = n. The operation ω is called nullary, unitary,

binary or ternary if its arity is 0, 1, 2, or 3 respectively. Occasionally we shall write ωA meaning the operation ω applied to An.

Remark 2.1 A nullary operation ω is determined by the image ω(∅) of the (unique) element ∅ in A0_{. Hence, ω can be viewed as an element of A, which we will call a constant.}

Example 2.1 A group G is an universal algebra (G, Ω) whose set of operation is Ω = {·, −1, 1}. Here, ω· = “ · ” is the binary operation of G (ρω· = 2) and ω−1 = −1 is the

operation which give us an inverse element x−1_{, ∀ x ∈ G, so ρ}

as being the constant (or nullary operation) of G, so ω1 = 1 is the neutral element of G

and ρω1 = 0. In addition a group satises the follows identities:

1. x · (y · z) = (x · y) · z 2. x · 1 = 1 · x = x 3. x · x−1 _{= x}−1_{· x = 1}

∀ x, y and z ∈ G. If G satises x · y = y · x, then G is called commutative (or abelian). Let S be a set. Let θ ⊂ S × S = S2 _{be a binary relation on S. We will denote the fact}

that (x, y) ∈ θ, where x, y ∈ S, by xθy.

Denition 2.3 The binary relation θ is an equivalence relation (e.r.) on S if, for any a, b and c ∈ S, the following holds:

i) aθa (reexivity)

ii) aθb ⇒ bθa (symmetry) iii) aθb, bθc ⇒ aθc (transitivity).

Denition 2.4 Let s ∈ S and θ be an e.r. on S. The equivalence class (or coset) of s modulo θ is the set s/θ = {x ∈ S : (x, s) ∈ θ} = [s]θ (sometimes denoted by s). Denote by

S/θ = {s/θ : s ∈ S} = {[s]_θ : s ∈ S} the set of all cosets modulo θ on S.

Proposition 2.1 Let θ an e.r. on S. We have i) S = Sx∈Sx/θ.

ii) for all x, y ∈ S, x/θ 6= y/θ ⇒ x/θ ∩ y/θ = ∅.

Proof. Note that for all x ∈ S we have that x ∈ x/θ by reexivity of θ, so S ⊂ Sx∈Sx/θ.

It's clear that x/θ ⊂ S for each x ∈ S, so Sx∈Sx/θ ⊂ S. So we have i). For item ii), note

that if there exists s ∈ x/θ ∩ y/θ then by transitivity x/θ = y/θ.

Denition 2.5 We say that B is a subalgebra of A if B ⊂ A and (B, Ω) is an algebra (denoted B ≤ A).

This means that B is closed with respect to all operations of A. Namely for all b1, . . . , bn ∈

B the inclusion ωA(b1, . . . , bn) ∈ B holds for all ω ∈ Ω, such that ρω = n.

Proposition 2.2 Let Ai ≤ A, i ∈ I. Then T_i∈IAi ≤ A.

Proof. Let ω ∈ Ω, ρω = n, and a1, . . . , an ∈

T

i∈IAi. So ai ∈ Aj, i = 1 . . . n, j ∈ I.

Then ω(a1, . . . , an) ∈ Aj ∀j ∈ I. That is ω(a1, . . . , an) ∈

T

i∈IAi and as Ti∈IAi ⊆ A,

concluding the proof.

Denition 2.6 Let (A, Ω) be an algebra and let θ an e.r. on A. We say that θ is a congruence on A if it satises: ∀ω ∈ Ω, ρω = n, and elements xi, yi ∈ A, if xiθyi holds

for i = 1, . . . n then

ωA(x1, . . . , xn)θωA(y1, . . . , yn)

.

Proposition 2.3 Let θi (i ∈ I) be a congruence on (A, Ω). Then θ := T_i∈Iθi is a

con-gruence on (A, Ω).

Proof. Its follows by the fact that θ ⊆ θi ∀i ∈ I.

Denition 2.7 Let (A, Ω) be an algebra and θ a congruence on A. The set A/θ is an algebra with signature Ω. We dene ∀ω ∈ Ω , ρω = n, and a1, . . . , an ∈ A

ωA/θ([a1]θ, . . . , [an]θ) =ω A

(a1, . . . , an)



θ

We say that (A/θ, Ω) is the quotient algebra A with respect to θ. The map from A to A/θ dened by a 7→ [a]θ is called natural map.

By Proposition 2.1, a natural map is a surjective map.

Denition 2.8 Let A, B be algebras under the same signature Ω. Let ϕ be a function from A to B and if for all ω ∈ Ω, ρω = n, and a1, . . . , an∈ A

ϕ(ωA(a1, . . . , an)) = ωB(ϕ(ai), . . . , ϕ(an)). (2.1)

holds then we say that ϕ is a homomorphism. An injetctive homomorphism is called monomorphism. If B is the image of ϕ and (2.1) holds then we say that ϕ is a epi-morphism. If ϕ is monomorphism and epimorphism then it is called isoepi-morphism.

If ϕ : A → B and ψ : B → C are homomorphisms. Then ψ ◦ ϕ : A → C is a homomorphism.

Let θ be a congruence on (A, Ω). Let νθ be the natual map, so for any ω ∈ Ω we have

that νθ(ωA(a1, . . . , an)) = [ω(a1, . . . , an)]θ = ωA/θ([a1]θ, . . . , [an]θ) = ωA/θ(νθ(a1), · · · , νθ(an)),

namely νθ is an epimosphism. We call νθ the natural epimorphism.

Denition 2.9 Let ϕ : A → B be a homomorphism. The set Kerϕ = {(a1, a2) ∈ A : ϕ(a1) = ϕ(a2)} ⊂ A2

is the kernel of ϕ.

Proposition 2.4 Let A, B be algebras under the same signature Ω and let ϕ : A → B be a homomorphism.

1) im(ϕ) ≤ B;

2) If ϕ is a monomorphism, then ϕ : A → im(ϕ) is isomorphism; 3) Kerϕ is a congruence.

Proof. Let ω ∈ Ω, ρω = n, and k1, . . . , kn∈im(ϕ), so there exist a1, . . . , an∈ Asuch that

ϕ(ai) = ki (i = 1, . . . , n). So ωB(k1, . . . , kn) = ωB(ϕ(a1), . . . , ϕ(an)) = ϕ(ωA(a1, . . . , an)).

As ωA_(a

1, . . . , an) ∈ A, we have that (im(ϕ), Ω) is an algebra, proving item 1. The

se-cond item follows because ϕ : A → im(ϕ) is an epimorphism. For 3) denote Kerϕ = θ. Note that for any ω ∈ Ω, ρω = n, if liθgi (i = 1, . . . , n), that is, (li, gi) belongs to θ =

{(a1, a2) ∈ A : ϕ(a1) = ϕ(a2)} (i = 1, . . . , n), then ω(ϕ(l1), . . . , ϕ(ln)) = ω(ϕ(g1), . . . , ϕ(gn)),

and hence ϕ(ω(l1, . . . , ln)) = ϕ(ω(g1, . . . , gn)) because ϕ is an homomorsm, so we have

that ω(l1, . . . , ln)θω(g1, . . . , gn). It is easy to see that θ is an e.r., completing the proof.

Let Hom(A, B) be the set of homomorphisms from A to B. By Propositions 2.3 and 2.4 the set

\

ϕ∈Hom(A,B)

Kerϕ

is a congruence on A.

Lemma 2.1 Let ϕ : A → B be a homomorphism, and let θ1 ⊆ A × A and θ2 ⊆ B × B be

congruences. If ϕ(θ1) = {(ϕ(x1), ϕ(x2)) : x1θ1x2} ⊆ θ2 then there exists a homomorphism

ψ : A/θ1 → B/θ2 such that ψη1 = η2ϕ, where η1 : A → A/θ1 and η2 : B → B/θ2 are the

Proof. Note that ϕ(θ1) is a congruence on ϕ(A) ≤ B. By the hypotesis we have that

ϕ(θ1) ⊆ θ2, so ϕ(θ1) ∩ θ2 = ϕ(θ1) and it follows that

ϕ(A)/ϕ(θ1) ∩ θ2 = ϕ(A)/ϕ(θ1) ≤ B/θ2.

Now, given x ∈ A/θ1 there exists a ∈ A such that η1(a) = x and so a corresponds to

the coset [ϕ(a)]ϕ(θ1). We apply the inclusion monomorphism ι([ϕ(a)]ϕ(θ1)) = [ϕ(a)]ϕ(θ1)

from ϕ(A)/ϕ(θ1) to B/θ2. The mapping ψ : x 7→ [ϕ(a)]ϕ(θ1) (η1(a) = [a]θ1 = x) is a

homomorphim because η2 ◦ ϕ = ψ ◦ η1 is a homomorphism. We obtain the commuting

diagram A/θ1 ψ _// ϕ(A)/ϕ(θ1) ι _// B/θ2 A η1 OO ϕ _// ϕ(A) η2 OO

which can be reduced to

A/θ1 ψ _// B/θ2 A η1 OO ϕ _// B η2 OO

Theorem 2.2 (Isomorphism Theorem). Let A, B be algebras with signature Ω and sup-pose ϕ : A −→ B is an epimorphism. Then there is an isomorphism ψ from A/ker(ϕ) to B dened by ϕ = ψ ◦ η, where η is the natural homomorphism from A to A/ker(ϕ).

A ϕ // η  B A/ker(ϕ) ψ ::

Proof. We have that ϕ(A) = B. If we replace θ1 by ker(ϕ) and ϕ(θ1) by ϕ(ker(ϕ)) in

Lemma 2.1, and observe that B/ϕ(ker(ϕ)) ∼= B, we conclud that there exists a homo-morphism ψ from A/ker(ϕ) to B such that τ ◦ϕ = ψ◦η, where τ is the identity map. Thus, we can identify B with B/ϕ(ker(ϕ)) and ψ([a]ker(ϕ)) = ϕ(a), thus ψ is an epimosphism.

If ϕ(a) = ϕ(b), we have that (a, b) ∈ Ker(ϕ) so [a]ker(ϕ) = [b]ker(ϕ), that is, ψ is injective,

i.e., an isomorphism. Indeed, obtain the following commuting diagram: A/ker(ϕ)ψ //B A η OO ϕ _// B τ =idB OO

completing the proof.

Denition 2.10 Let A be an algebra on Ω, X ⊂ A and BX = {B ≤ A; X ⊂ B}

the minimal element of BX is called the subalgebra generated by the set X. We denote

this subalgebra by hXi.

Proposition 2.5 hXi = TB∈BXB.

Proof. By Proposition 2.2, TB∈BXB ≤ A. By Denition 2.10, we have that hXi ⊆ B

for all B ∈ BX, so hXi ⊆ TB∈BXB. Since hXi ∈ BX, we have TB∈BXB ⊆ hXi. Thus,

hXi =T

B∈BXB.

Let Ω be a signature and X 6= ∅. We dene

M0(X) = {ω; ω ∈ Ω and ρω = 0} ∪ X

and for all ω ∈ Ω, ρω = n > 0, we dene Mi+1(X) by

( ωm1, . . . , mn: mk∈ i [ j=0 Mj(X), k = 1, . . . , n, ∃ i0; 1 ≤ i0 ≤ n; mi0 ∈ Mi(X) )

It is clear that Mi(X) 6= ∅ ∀ i. The algebra

M (X) =

∞

[

i=0

Mi(X)

with signature Ω so that ω(m1, . . . , mn) := ωm1. . . mn, is called algebra of terms on

X.

Proposition 2.6 M(X) has the following propeties: (i) If i 6= j then Mi(X) ∩ Mj(X) = ∅;

(ii) M(X) = hXi .

Proof. (i) If i > 0, note that Mi(X) ∩ X = ∅. If x ∈ M0(X) ∩ Mj(X) for some j > 0

then x = ω(m1, . . . , mn), ρω = n > 0. But also x ∈ M0(X), then ρω = 0, it is a

i < j). Let ω1(m1, . . . , mn1) ∈ Mj+1(X), ω2(l1, . . . , ln2) ∈ Mi(X) and ω1(m1, . . . , mn1) =

ω2(l1, . . . , ln2), so ω1 = ω2, n1 = n2 = n and mk = lk (1 ≤ k ≤ n). By denition of

Mj+1(X), there exist k1, 1 ≤ k1 ≤ n, such that mk1 ∈ Mj(X). By denition of Mi(X),

there exists k0 < i such that lk1 ∈ Mk0(X). So mk1 = lk1 and Mk0(X) ∩ Mj(X) 6=

∅. Repeating a nite number of steps, we will have M0(X) ∩ Mt(X) 6= ∅, (0 < t), a

contradiction. So, the proof follows by induction.

(ii) It is cleat that hXi ⊆ M(X). If B ∈ BX, B is an algebra on signature Ω and

{ω ∈ Ω; ρω = 0} ⊂ B, so M0(X) ⊂ B. Now suppose that Mi(X) ⊂ B, 1 ≤ i ≤ k. If

m1, . . . , mn ∈

Sk

j=0Mj(X), we have that ω(m1, . . . , mn) ∈ B ∀ω ∈ Ω, ρω = n, because

Sk

j=0Mj(X) ⊂ B. So, Mk+1(X) ⊂ B and by induction M(X) ⊂ B. Then, for any B ∈ BX

we have that M(X) ⊂ B, that is M(X) ⊆ TB∈BXB = hXi.

Example 2.3 Let Ω = {+, ·} be the signature with two binary operation simbols, and let X = {x, y, z}. Then

x, y, z,

+(x, y) = x + y, · (x, y) = x · y, and + (·(x, z), ·(x, y)) = (x · z) + (x · y) are some of the terms over X.

2.2 Variety of algebras

Let X = {x1, . . . , xn} be a set of variables (alphabet). Let Hom(M(X), A) be the set

of homomorphisms ϕ : M(X) → A

Denition 2.11 Let f(x1, . . . , xn), g(x1, . . . , xn) ∈ M (X). We say that an algebra A

fullls the identity

f (x1, . . . , xn) = g(x1, . . . , xn)

if for every ϕ ∈ Hom(M(X), A) we have

ϕ(f (x1, . . . , xn)) = ϕ(g(x1, . . . , xn)).

Denition 2.12 Let I be a set of identities. The variety dened by I is the class of all algebras with the same signature Ω such that the identities of I fullll. We write Var(I). Remark 2.2 Let I1, I2 be two sets of identities. If I1 ⊆ I2, then Var(I2) ⊆Var(I1).

We consider the algebra H with signature Ω. The set of all identities which fulll in algebra H we denote by I (H) ⊂ M (X) × M (X). The variety V ar (I (H)) is called a variety generated by algebra H. We denote for short V ar (I (H)) = V ar (H).

Let Θ be a variety of algebras with signaute Ω. Let F ∈ Θ and X ⊂ F .

Denition 2.13 We say that F is a free algebra of the variety Θ generated by the set X of free generators if for any A ∈ Θ and for every mapping s : X → A there exist a unique homomorphism ϕ : F → A such that ϕ|X = s. We denote FΘ(X) = F.

Theorem 2.4 Suppose that F (X1) and F (X2) are two generated free algebras of Θ and

|X1| = |X2|, then F (X1) ∼= F (X2).

Proof. Note that the identity map

ik: Xk→ Xk k = 1, 2,

has an unique extension up to a homomorphism from F (Xk)to F (Xk): the identity map.

Now let

ψ : X1 → X2

be a bijection. Then we have a homomorphism

ϕ : F (X1) → F (X2)

extending ψ, and a homomorphism

γ : F (X2) → F (X1)

extending ψ−1_{. As ϕ ◦ γ is an homomorphism extending i}

2, it follows by Denition 2.13

that ϕ ◦ γ is the identity map on F (X2). Likewise γ ◦ ϕ is the identity map on F (X1).

Thus ϕ is a bijection, so F (X1) ∼= F (X2).

Example 2.5 Let X = {xi | i ∈ I} and dene formally the set X−1 = x−1i | i ∈ I

. A word in the alphabet X is a nite sequence of symbols from X ∪ X−1_{. We denote the}

empty word by υ. A word is called reduced if it does not contain subwords of the form xα_ix−α_i , α = ±1, and we will delete these subwords whenever possible. Two words w1, w2

are called equivalent, if w2 can be obtained from w1 by a nite number of insertions and

deletions of words of the form xα ix

−α

i . This relation is an e.r. on the set of all words, and

set F (X) = {[w] | w word in alphabet X} we dene [w1] [w2] = [w1w2] (this multiplication

is well dened). So, F (X) with this multiplication is a semigroup. In fact, it is a group, as wee show below.

(i) Let ρ(w) be the reduced word obtained from w by successive deletions, of subwords of the form xα

ix −α

i ,moving from right to left. The function ρ fullls

ρ(w1w2) = ρ(w1ρ(w2)) (2.2) ρ(xα_ix−α_i w) = ρ(w) (2.3) ρ(w1xαix −α i w2) = ρ(w1w2) (2.4) ρ(w1w2) = ρ(ρ(w1)ρ(w2)), (2.5)

α = ±1. Property (2.2) is immediate from the denition of ρ; (2.3) follows from (2.2); (2.4) follows from (2.2) and (2.3); and nally (2.5) follows from (2.2), (2.3), (2.4) by induction on the length of w1. Suppose now that w1 is equivalent to w2,

where w1, w2 are reduced words. By denition there is a sequence

w1 ≡ u1, u2, . . . , un ≡ w2

of words, such that neighboring words dier by a single subword of the form xα ix

−α i .

Therefore by (2.4) we have that ρ(uj) ≡ ρ(uj+1), whence ρ(w1) = ρ(w2). Since w1

and w2 are reduced this implies that w1 = w2.

(ii) We conclud that the product [w1] [w2] is independent of the choice of representatives

[w1], [w2], by (i) and (2.5). The associativity of the operation follow by denition.

The identity element is the equivalence class containing the empty word, and the inverse of the class xα1

i1 · · · x αn in  is the class x−αn n1 · · · x −α1 i1  .

The group F (X) is called the free group freely generated by the set X, and the cardinal number |X| the rank of the free group.

Theorem 2.6 Suppose a group G is generated by a set V = {vj|j ∈ I}, and let X be

an alphabet {xi|i ∈ I} . The map f : X → V dened by f(xi) = vi, extends to a unique

epimorphism F (X) → V .

Proof. We dene the image of the class xα1

i1 · · · x

αn

in



under f as the element vα1

i1 · · · v

αn

in .

It follows directly from denition that this denes a map f from F (X) onto G, and that this map is a homomorphism.

As a example, consider the subvariety Θ of the variety of groups, dened by the identities x4 = 1, (2.6) ((x1, x2), (x3, x4)) = 1, (2.7) and ((((x1, x2), x3), x4), x5) = 1, (2.8)

in other words, this is a variety of metabelian Sanov (SANOV, 1940) groups which are nilpotent of class ≤ 4. This subvariety plays a central role in this thesis. We will use the method of verbal operations elaborated in (PLOTKIN; ZHITOMIRSKI, 2006) for the

calculation of the quotient group A/Y for the variety Θ and will prove that this group is not trivial. The method of verbal operations we will explain in the next chapter.

2.3 Category

In discussing algebraic geometry we need to undertand the formalism of categories. Briey, a category K consist of

Denition 2.14 A pair (ObK,MorK) = K, such that ObK is the set of objects and MorK

is the set of arrows (or morphisms) between objects, and K fullls

i) Let A, B, C ∈ ObK and f, g ∈ MorK, f : A → B, g : B → C so, there exist

g ◦ f : A → C ∈MorK;

ii) ∀ A ∈ ObK there exists 1A: A → A (1A∈MorK) , such that if f : A → B

f ◦ 1A = f and 1B◦ f = f ;

iii) ∀ f : A → B, g : B → C ∈ MorK

f ◦ (g ◦ h) = (f ◦ g) ◦ h is fullled ∀ h : C → D, A, B, C, D ∈ ObK;

iv) Let HomK(A, B) be the set of all arrows from A to B. If the ordered pair

HomK(A, B) ∩HomK(C, D) = ∅.

Example 2.7 The category Grp of groups, whose objects are groups and whose maps are group homomorphisms.

We also dene the concept of functor. A covariant functor is a morphism of categories. Denition 2.15 Let K1 and K2 be two categories. A covariante functor Φ : K1 → K2 is a

pair Φ = (ΦO, ΦM) of functions such that ΦO :ObK1 →ObK2 and ΦM acts on morphisms

ΦM :MorK1 →MorK2. If ϕ1 ∈HomK1(A, B), ϕ2 ∈HomK1(B, C) so we have that

Φ(ϕ2◦ ϕ1) = Φ(ϕ2) ◦ Φ(ϕ1) ∈HomK2(Φ(A), Φ(C)).

For all A ∈ K, Φ(1A) = 1Φ(A).

We consider in this thesis only covariant functors, which now we shall refer to as just functors. Let A, B ∈ ObK1 and ϕ ∈ HomK1(A, B). The following diagram represents

Denition 2.15. A ϕ // ΦO  B ΦO  Φ(A) ΦM(ϕ) //_Φ(B)

Let K1, K2 and K3 be categories. Let Φ1 : K1 → K2 and Φ2 : K2 → K3 functors. The

composite functions

A 7→ Φ2(Φ1(A)) ϕ 7→ Φ2(Φ1(ϕ))

on objects A and maps ϕ of K1 dene a functor Φ2◦ Φ1 : K1 → K3 called the composite

of Φ2 with Φ1. This composition is associative. For each category K there is an identity

functor IK : K → K which acts as an identity for this composition. An automorphisms

of K is a functors Φ1 such that there exists a functor Φ2 : K → K with Φ1◦ Φ2 = IK. We

write Φ2 = Φ−11 . The set of all automorphisms A of K is a group.

Example 2.8 For any group G the set of all products of commutators (x, y) = x−1_y−1_xy,

(x, y ∈ G), is a normal subgroup (G, G) of G, called the commutator subgroup. Since any homomorphism G → H of groups carries commutators to commutators, the assignment G → (G, G) denes a functor Grp → Grp.

Example 2.9 Let Ab be the category of all abelian groups. We dene the functor Π from Grp to Ab by Π(G) = G/(G, G) ∀G; and for any ϕ ∈ MorGrp(G, H)we dene Π(ϕ) = ϕ,

where for all g ∈ G, ϕ(g) = h, ϕ(g) = ϕ(g) = h ∈ H/(H, H). The denition of ϕ is correct because ϕ ((G, G)) ≤ (H, H).

Denition 2.16 Let A and B be categories and let Ψ1, Ψ2 : A → B be functors. A

natural transformation α : Ψ1 → Ψ2 is a family ( Ψ1(A) αA //

Ψ2(A) )A∈ObA of maps

in B such that for every morphism A f _//

A0 in K1 the diagram Ψ1(A) Ψ1(f )_// αA  Ψ1(A0) α_A0  Ψ2(A) _Ψ 2(f ) //_Ψ2(A0₎

commutes. The maps αA are called the components of α.

Remark 2.3 A subcategory C of a category K is a collection of some of the objects and some of the arrows of K, which includs with each arrow ϕ : A → B both the object A and the object B, with each object A its identity arrow 1A and with each pair of composable

arrows A → B → C their composite. It follows that the inclusion map C → K which sends each object and each arrow of C to itself in K, is a functor (the inclusion functor). Example 2.10 For each group G the projection ρG : G → G/(G, G) to the

factor-commutator group denes a transformation ρ from the identity functor on Grp to the functor Φ = Γ ◦ Π, Grp Π _//Ab Γ _{//Grp , where Γ is the inclusion functor.}

More-over, the transformation ρ : Φ → IGrp is natural, because each group homomorphism

φ : G1 → G2 denes a homomorphism Γ(Π(φ)) = φ0, such that φ0(g1) = φ(g1) ∈ H/(H, H)

and for which the following diagram commutes I_Grp(G1) = G1 φ _// ρ_G1  G2 = IGrp(G2) ρ_G2  Γ(Π(G1)) = G1/(G1, G1) φ0 //_{G2/(G2, G2) = Γ(Π(G2))}

Denition 2.17 We say that functors Ψ1 and Ψ2 are naturally isomorphic (denoted Ψ1 ∼=

Ψ2) if the components of the natural transformation are isomorphisms.

Remark 2.4 Two categories A, B are equivalent if there exist two functors Ψ1 : A → B, Ψ2 : B → A such that Ψ1◦ Ψ2 ∼= IB and Ψ2◦ Ψ1 ∼= IA.

3 The method of verbal operations

In this section we will explain the method of verbal operations for the computa-tion of the quotient group A/Y in the case of arbitrary variety Θ of universal algebras of signature Ω. The reader can nd the explanation and application of this method in ( PLOT-KIN; ZHITOMIRSKI, 2006), (TSURKOV, 2007a), (TSURKOV, 2007b), (TSURKOV, 2016) and

(TSURKOV, 2017).

3.1 First denitions and basic facts

This method applies only if the following condition holds in the variety Θ.

Condition 3.1 (PLOTKIN; ZHITOMIRSKI, 2006) Φ(F (x)) ∼= F (x)for every automorphism Φ of the category Θ0 and for every x ∈ X

0.

In this case, by (PLOTKIN; ZHITOMIRSKI, 2006), for every Φ ∈ A there exists a system

of bijections

S = sF : F → Φ(F ) | F ∈ ObΘ0 , (3.1)

such that for every ψ ∈ MorΘ0(A, B) the diagram

A sA// ψ  Φ(A) Φ(ψ)  B _s B //_Φ(B)

is commutative. In other hands Φ acts on the morphisms ψ : A → B of Θ0 as follows:

Φ(ψ) = sBψs−1A . (3.2)

Denition 3.1 We say that the system of bijections (3.1) is a system of bijections asso-ciated with the automorphism Φ ∈ A if this system fullls condition (3.2).

In generalan arbitrary automorphism of the category Θ0 _{can be associated with}

vari-ous systems of bijections and some system of bijections can be associated with varivari-ous automorphisms. Ithe paper (PLOTKIN; ZHITOMIRSKI, 2006) also consides special

autho-morphisms which are called strongly stable.

Denition 3.2 An automorphism Φ of the category Θ0 _{is called strongly stable if it}

sa-tises the conditions:

1. Φ preserves all objects of Θ0_,

2. there exists a system {sF} of bijections associated with the automorphism Φ such

that

sF |X= idX (3.3)

holds for every F (X) ∈ ObΘ0.

In other words, the system {sF} preserve all generators of domains. It is clear that the

set S of strongly stable automorphisms of the category Θ0 _{is a subgroup of the group A}

of automorphisms of this category. By (PLOTKIN; ZHITOMIRSKI, 2006), A = YS holds if

the category Θ0 _{fullls Condition 3.1. In this case we have that A/Y ∼= S/S ∩ Y. So the}

study of A/Y may be reduced to the study of the groups S and S ∩ Y.

3.2 Strongly stable automorphism and strongly stable

system of bijections

Consider a strongly stable automorphism Φ ∈ S. There exists a system of bijections associated with this automorphism which satises Denition 3.2. This system of bijections is uniquely dened by the automorphism Φ, because the equality sA(a) = Φ(α)(x) holds

for every A ∈ ObΘ0 _{and every a ∈ A, where α : F (x) → A is a homomorphism dened}

by α(x) = a (see (TSURKOV, 2016)). We denote this system of bijections by SΦ, and its

bijections by sΦ

F (F ∈ ObΘ0).

Denition 3.3 The system of bijections S = {sF : F → F | F ∈ObΘ0} is called

stron-gly stable if for every A, B ∈ ObΘ0 _{and every µ ∈ Mor}

Θ0(A, B) the mappings s_Bµs−1_A ,

The set of strongly stable system of bijections is denoted by SSSB. It is clear that the system of bijections SΦ is strongly stable. Hence the mapping A : S → SSSB such that

A(Φ) = SΦ is well dened by (TSURKOV, 2016). This mapping is one to one and onto by

(TSURKOV, 2016). If Φ1, Φ2 ∈ S then there are strongly stable systems of bijections

A(Φ1) = SΦ1 =s Φ1 F : F → F | F ∈ObΘ 0 and A(Φ2) = SΦ2 =s Φ2 F : F → F | F ∈ObΘ 0

For every ϕ ∈ MorΘ0(F₁, F₂)we have the equality Φ₁Φ₂(ϕ) = sΦ_F2 2s Φ1 F 2ϕ s Φ1 F1
−1 sΦ2 F1
−1 . It means that the system of bijections

sΦ2

F s Φ1

F : F → F | F ∈ObΘ 0

is associated with the automorphism Φ1Φ2. But it is clear that this system is strongly

stable, so this system of bijections is a uniquely dened strongly stable system of bijections corresponds to the strongly stable automorphism Φ1Φ2, in other words,

A(Φ1Φ2) = sΦF2s Φ1

F : F → F | F ∈ObΘ 0 _.

3.3 Strongly stable system of bijections and applicable

systems of words

We consider an algebra F = F (x1, . . . , xn) ∈ ObΘ0 and take a word (element) w =

w(x1, . . . , xn) ∈ F (x1, . . . , xn).

Denition 3.4 The operation ω∗ _{: ω}∗_(h

1, . . . , hn) = w(h1, . . . , hn) is called verbal

opera-tion dened on the algebra H by the word w, where hi ∈ H, 1 ≤ i ≤ n, and H ∈ Θ is an

arbitrary algebra of the variety Θ.

The reader can compare this denition with the denition of word maps, (SEGAL, 2009), (KANEL-BELOV; KUNYAVSKII; PLOTKIN, 2013) and references therein.

ρω we consider the algebra Fω = F (x1, . . . , xρω) ∈ ObΘ

0_{. Having a system of words}

W = {wω | ω ∈ Ω} where wω ∈ Fω, denote by HW∗ the algebra which coincides with H

as a set, but instead of the original operations {ω | ω ∈ Ω} it possesses the system of operations {ω∗ _{| ω ∈ Ω} where ω}∗ _{is a verbal operation dened by word w}

ω.

We can consider algebras H and H∗

W with the same signature Ω: the realization of the

operation ω ∈ Ω in the algebra H is the operation ω and the realization of the operation ω ∈ Ω in the algebra H_W∗ is the operation ω∗. So, if A and B are algebras with original operations {ω | ω ∈ Ω}, A∗

W and B ∗

W are algebras with operations {ω

∗ _{| ω ∈ Ω}, we can}

consider homomorphisms from A to B∗

W, from A ∗

W to B and so on.

Denition 3.5 The system of words W = {wω | ω ∈ Ω} is called applicable if

wω(x1, . . . , . . . xρω) ∈ Fω

and for every F = F (X) ∈ ObΘ0 _{there exists an isomorphism s}

F : F → FW∗ such that

sF |X= idX.

The set of all applicable systems of words we denote by ASW. This set is never empty. The trivial example of a applicable system of words, which always exists, is the system W = {wω | ω ∈ Ω}, such that wω = ω for every ω ∈ Ω.

We suppose that W = {wω | ω ∈ Ω} is an applicable system of words and consider

the system of isomorphisms S = {sF : F → FW∗ | F ∈ObΘ0} mentioned in Denition

3.5. The isomorphism sF as a mapping from the algebra F ∈ ObΘ0 to itself is only a

bijection, which fulll conditions (3.3). The mappings sBµs−1A , s −1

B µsA : A → B are

ho-momorphisms by (TSURKOV, 2016) for every A, B ∈ ObΘ0and every µ ∈ Mor

Θ0(A, B). So

S = {sF : F → F | F ∈ObΘ0}is a strongly stable system of bijections. From (TSURKOV,

2016) we conclude that the isomorphisms sF : F → FW∗ such that (3.3) holds are uniquely

dened by the system of words W . So the system of bijections S is uniquely dened by W. We denote this system by SW. Therefore the mapping B : ASW → SSSB such that

B(W ) = SW is well dened. This mapping is one to one and onto by (TSURKOV, 2016).

In particular, if system of bijections S = {sF : F → F | F ∈ObΘ0} is a strongly stable

system of bijections, then a word wω from the applicable system of words W = B−1(S)

we can obtain by the formula

wω(x1, . . . , xρω) = sFω(ω(x1, . . . , xρω)) ∈ Fω,

conclude (TSURKOV, 2016) that there is one to one and onto correspondence C = B−1_{A :}

S→ ASW. We denote C(Φ) by WΦ. The systems of words WΦis dened by formula (3.3)

where bijections sFω = s

Φ

Fω are the corresponding bijections of the system A(Φ) = SΦ.

Therefore we can calculate the group S if we are able to nd all applicable system of words. If Φ1, Φ2 ∈ Sand A(Φ1) = SΦ1 =s Φ1 F : F → F | F ∈ObΘ 0 _, A(Φ2) = SΦ2 =s Φ2 F : F → F | F ∈ObΘ 0

are strongly stable systems of bijections corresponded to automorphisms Φ1 and Φ2, then

as we saw in the previous section, the strongly stable system of bijections A(Φ2Φ1) = S =sΦF2s

Φ1

F : F → F | F ∈ObΘ 0

corresponds to the strongly stable automorphism Φ2Φ1. Hence, by (3.3), the applicable

systems of words B−1_{(S) = C(Φ}

2Φ1) we can obtain by formula

wω(x1, . . . , xρω) = s Φ2 Fωs Φ1 Fω(ω(x1, . . . , xρω)) ∈ Fω, where ω ∈ Ω.

3.4 Automorphisms, which are strongly stable and

in-ner

For calculation of the group S ∩ Y we also have the following

Criterion 3.1 (PLOTKIN; ZHITOMIRSKI, 2006, Lemma 3)The strongly stable automorphism

Φ of the category Θ0_{, such that C (Φ) = W}

Φ = W, is inner if and only if for every

F ∈ ObΘ0 there exists an isomorphism cF : F → FW∗ such that

cBψ = ψcA (3.4)

is fullled for every A, B ∈ ObΘ0 and every ψ ∈ Mor

Θ0(A, B).

Also we have

Proposition 3.1 (GOMES, 2017, Proposition 23) The system of functions

MorΘ0(A, B) if and only if there exists c(x) ∈ F (x) such that

cA(a) = c(a), (3.5)

4 Application of the method of

verbal operations.

We consider every group as universal algebra with signature which has 3 operations: Ω = {1, −1, ·} ,

where the 0-ary operation 1 gives us an unit of a group, the 1-ary operation −1 gives us for an arbitrary element g of a group G the inverse element g−1 _{and the 2-ary operation}

“ · ”gives us the product of two emlements of a group G.

The IBN (invariant basis number) property or invariant dimension property was de-ned initially in the theory of rings and modules, see, for example, (HUNGERFORD, 1974,

Denition 2.8). But then this concept was generalized to arbitrary varieties of algebras: Denition 4.1 We say that the variety Θ has an IBN property if for every FΘ(X) ,

FΘ(Y ) ∈ ObΘ0 the FΘ(X) ∼= FΘ(Y ) holds if and only if |X| = |Y |.

From here on Θ is a subveriety of the variety of groups dened by identities (2.6), (2.7) and (2.8). By (FUJIWARA, 1955) our variety Θ has an IBN property. It is easy to conclude

from this fact that in the variety Θ Condition 3.1 fullls. From here on Θ is a subveriety of the variety of groups dened by identities (2.6), (2.7) and (2.8). So, the method of verbal operations is valid in our variety.

Thus the strategy of our research is clear. First of all we will understand the structure of the 2-generated free group FΘ(x, y)of our variety.

After that we will nd all applicable system of words

W = {w1, w−1(x) , w·(x, y)} , (4.1)

where w1 is a constant which corresponds to the 0-ary operation 1, w−1(x) ∈ FΘ(x) is

which corresponds to the 2-ary operation ·. We will use Denition 3.5 for the nding of the applicable system of words. The necessary conditions for the system of words to be applicable we will conclude from the fact that the isomorphism sF : F → FW∗ , which

exists for every F ∈ ObΘ0_{, provide the fullling of all identities of the variety Θ in the}

groups F∗

W. It will give us 4 systems of words of the form (4.1), which can be applicable.

In the next step of our research we will prove that all these systems of words are applicable. We will prove that for all these system W all identities of the variety Θ really fulll in the groups F∗

W for every F ∈ ObΘ0. This will allow us to construct

the homomorphism s = sF (X) : F (X) → (F (X)) ∗

W, such that s|X = idX for every

F (X) ∈ ObΘ0_{. After that we will nd the inverse maps for every s}

F (X). It allow to

conclude that all homomorphisms sF (X) are isomorphisms and all 4 considered systems of

words are applicable and provide the strongly stable automorphisms of the category Θ0.

And we will nish our research when we will compute for the category Θ0 _{the group}

Y∩ Sby Criterion 3.1. We will see at the end of our research that the group A/Y of the category Θ0 _{contains 2 elements.}

5 Some properties of the varieties

N

₄

and Θ

In this paper N4 is the variety of nilpotent groups of class no more then 4. The free

groups of this variety, generated by x1, . . . , xn we will denote by N4(x1, . . . , xn).

We will write ((((x, y) , z) , . . .) , t) as (x, y, z, . . . , t). Also we will denote γ1(G) = G

and γi+1(G) = (γi(G) , G) for every group G. And we will write by Z (G) the center of

the group G.

In our computation we will frequently use the identities

(xy, z) = (x, z)y(y, z) = (x, z)(x, z, y)(y, z), (5.1)

(x, yz) = (x, z)(x, y)z = (x, z)(x, y)(x, y, z), (5.2) (x−1, y) = (y, x)x−1 = (x, y)−1(y, x, x−1) (5.3) which fulll in every group (see (HALL, 1959, (10.2.1.2) and (10.2.1.3)) and ( KARGAPO-LOV; MERZLJAKOV, 1979, p. 20, (3))). From these identities we can conclude these facts

about an arbitrary group G ∈ N4:

1. for every g1, g2 ∈ G and every l1, l2 ∈ γ4(G)

(g1l1, g2l2) = (g1, g2) , (5.4)

2. for every g ∈ G and every l1, l2 ∈ γ2(G)

(l1l2, g) = (l1, g) (l2, g) , (g, l1l2) = (g, l2) (g, l1) , (5.5)

3. for every g1, g2, g3 ∈ Gand every l1, l2, l3 ∈ γ3(G)

4. for every g1, g2, g3, g4 ∈ G and every l1, l2, l3, l4 ∈ γ2(G)

(g1l1, g2l2, g3l3, g4l4) = (g1, g2, g3, g4) , (5.7)

5. every commutator of the length 4 is a multiplicative function by all its 4 arguments: w (g1, . . . , gili, . . . , g4) = w (g1, . . . , gi, . . . , g4) w (g1, . . . , li, . . . , g4) , (5.8)

where w (x1, . . . , x4) ∈ γ4(N4(x1, . . . , x4)), 1 ≤ i ≤ 4, holds for every g1, . . . , g4, li ∈

G.

For every G ∈ N4 we have that γ4(G) ⊆ Z (G) and γ5(G) = {1}. For every G ∈ Θ

the group γ2(G) is an abelian group. We will use these facts later in our computations

without special reminder. Also we use the identity yx = xy(y, x), which fullls in every group, and the identity (2.6) which fullls in every group of the variety Θ without special reminder.

In this subsection we will describe the free group of our variety Θ generated by 2 generators. This group is a quotient group N4(x, y) /T, where T is a normal subgroup of

the identities with two variables of the subvariety Θ in the variety N4.

By (KARGAPOLOV; MERZLJAKOV, 1979, Theorem 17.2.2), if G is nitely generated

nilpotent group then there exist central (in particular, normal) series: 1 = G0 ≤ G = G1 ≤ G2 ≤ · · · ≤ Gs = G

such that Gi+1/Gi = hCiGi+1i (⇐⇒ Gi = hCi, Gi+1i), Ci ∈ Gi.

We have that hCiGi+1i ∼= Zn (n ≥ 2), or hCiGi+1i ∼= Z. Therefore every g ∈ G can be

uniquely represented in the form g = Cα1

1 C α2

2 . . . Csαs, where 0 ≤ αi < n, when hCiGi+1i ∼=

Zn, and αi ∈ Z, when hCiGi+1i ∼= Z.

Denition 5.1 We say that the set {C1, C2, ..., Cs}is a base of the group G and numbers

α1, α2, ..., αs are coordinates of the element g in this base.

The base of N4(x, y)we can denote by

C1 = x, C2 = y, C3 = (y, x), C4 = (y, x, y) , C5 = (y, x, x) , (5.9)

This is a base of Shirshov, which we can compute by the algorithm explained in ( BAHTU-RIN, 1987, 2.3.5).

In particular, if we substitute in (HALL, 1959, 10.2.1.4) (y, x) instead x and x instead

z, we obtain

(y, x) , y−1, x
y y, x−1, (y, x)
x x, (y, x)−1, y
(y,x) = 1. So, by (5.8), we can conclude that

((y, x) , y, x)−1(y, x, (y, x))−1(x, (y, x) , y)−1 = 1. We have that

(y, x, (y, x)) = ((y, x) , (y, x)) = 1, and

(x, (y, x) , y) = ((x, (y, x)) , y) = ((y, x) , x)−1, y
= (y, x, x, y)−1, hence

(y, x, y, x)−1(y, x, x, y) = 1 and

(y, x, y, x) = (y, x, x, y) = C8. (5.10)

Proposition 5.1 The identity

(xy)4 = x4y4(y, x)6(y, x, y)14(y, x, y, y)11(y, x, x)4(y, x, x, y)11(y, x, x, x), (5.11) fullls in the variety N4.

Proof. We will consider the group G ∈ N4 and x, y ∈ G.

Initially we are going to compute (xy)2_{. We have that}

(xy)2 = xyxy = x2y(y, x)y = x2y2(y, x)(y, x, y).

After this we compute (xy)3 _{by same method:}

(xy)3 = (xy)2(xy) = x2y2(y, x)(y, x, y)xy = x2y2x(y, x)(y, x, x)(y, x, y)(y, x, y, x)y. Now we will compute y2_x:

xy2(y, x)(y, x, y)(y, x) = xy2(y, x)2(y, x, y), (5.12) because elements of γ2(G)commute with elements of γ3(G) in every G ∈ N4. Hence, by

(5.12) and (5.10), we have the equality

(xy)3 = x3y2(y, x)3(y, x, y)2(y, x, x)(y, x, y, x)y =

x3y3(y, x)3(y, x, y)3(y, x, y)2(y, x, y, y)2(y, x, x)(y, x, x.y)(y, x, y, x)

⇒ (xy)3 _{= x}3_y3_{(y, x)}3_{(y, x, y)}5_{(y, x, x)(y, x, y, y)}2_{(y, x, x, y)}2_{. (5.13)}

Now we will compute (xy)4_{. By (5.13) we have that}

(xy)4 = xy(xy)3 = xyx3y3(y, x)3(y, x, y)5(y, x, x)(y, x, y, y)2(y, x, y, x)2 (5.14) After this we can compute that

yx3 = (yx) x2 = xy(y, x)x2 = x2y(y, x)2(y, x, x)x =

x3y (y, x)3(y, x, x)3(y, x, x, x) , therefore

xyx3y3 = x4y (y, x)3(y, x, x)3(y, x, x, x) y3. (5.15) We have that

(y, x, x, x) y3 = y3(y, x, x, x) . (5.16) Also we can compute that

(y, x, x)3y3 = y (y, x, x)3(y, x, x, y)3y2 =

y (y, x, x)3y2(y, x, x, y)3 = y2(y, x, x)3y (y, x, x, y)6

⇒ (y, x, x)3y3 = y3(y, x, x)3(y, x, x, y)9 (5.17) and

(y, x)3y3 = y (y, x)3(y, x, y)3y2 = y2(y, x)3(y, x, y)3(y, x, y)3(y, x, y, y)3y =

y2(y, x)3(y, x, y)6(y, x, y, y)3y =

y3(y, x)3(y, x.y)3(y, x, y)6(y, x, y, y)6(y, x, y, y)3 = y3(y, x)3(y, x.y)9(y, x, y, y)9

Therefore, by (5.15), (5.16), (5.17) and (5),

xyx3y3 = x4y4(y, x)3(y, x.y)9(y, x, y, y)9(y, x, x)3(y, x, x, y)9(y, x, x, x) . (5.19) After this, we have, by (5.14) and (5.19), that

(xy)4 = x4y4(y, x)3(y, x.y)9(y, x, y, y)9(y, x, x)3(y, x, x, y)9(y, x, x, x) ·

(y, x)3(y, x, y)5(y, x, x)(y, x, y, y)2(y, x, y, x)2 =

x4y4(y, x)6(y, x.y)14(y, x, y, y)11(y, x, x)4(y, x, x, y)11(y, x, x, x) .

By (5.10) we have Corollary 1 The identity

1 = (y, x)2(y, x, y)2(y, x, x, x)(y, x, y, y)−1(y, x, y, x)−1. (5.20) fullls in the variety Θ.

We denote the images of elements of the base {C1, . . . , C8} by the natural

homo-morphism N4(x, y) → N4(x, y) /T = FΘ(x, y) by the same notation: {C1, . . . , C8}.

Proposition 5.2 The relations:

C_i2 = 1, (4 ≤ i ≤ 8) (5.21)

C₃2C6C7C8 = 1 (5.22)

in FΘ(x, y) are consequences from the identities of Θ.

Proof. The equality (5.20) is an identity in Θ, so in (5.20) we can substitute x instead y and vice versa. Therefore

1 = (x, y)2(x, y, x)2(x, y, y, y)(x, y, x, x)−1(x, y, x, y)−1 =

(y, x)2(y, x, y)2(y, x, x, x)(y, x, y, y)−1(y, x, y, x)−1. (5.23)

By (5.3), (5.7), (5.8) and (5.10) we have that (y, x)2 _{= (x, y)}2_{, (y, x, x, x) = (x, y, x, x)}−1_,

that

(x, y, x)2(y, x, y, x) = (y, x, y)2(y, x, y, x)−1. (5.24) Also we have by (5.3), that

(x, y, x) = (y, x, x)−1(x, (y, x), (x, y)) = (y, x, x)−1 = C₅−1. Therefore (x, y, x)2 _{= C}−2

5 = C52. Now we conclude from (5.24) that

C₄2 = C₅2C₈2. (5.25)

Now we substitute in (5.20) (y, x) instead x and x instead y: 1 = (x, (y, x))2(x, (y, x), x)2(x, (y, x), (y, x), (y, x))·

(x, (y, x), x, x)−1(x, (y, x), x, (y, x))−1 = (x, (y, x))2(x, (y, x), x)2 = (y, x, x)−2(y, x, x, x)−2 So the relation

C₅2C₆2 = 1 (5.26) holds.

Analogously we substitute in (5.20) y instead x and (y, x) instead y and conclude that

1 = C₄2. (5.27)

Now by (5.25) and (5.26) we have that

C₅2 = C₆2 = C₈2. (5.28)

Also, when we substitute in (5.20) (y, x, x) instead y, we obtain that

1 = C₆2. (5.29)

And when we substitute in (5.20) (y, x, y) instead x, we conclude

1 = C₇−2 = C₇,2. (5.30) Therefore, we conclude (5.21) from (5.27), (5.28), (5.29), (5.30). And after this the equality (5.20) has form

1 = C₃2C₄2C6C7−1C −1

. This completes the proof.

Now we consider in N4(x, y) the minimal normal subgroup R which contains

ele-ments x4_{, y}4 _{and the left parts of the relations (5.21) and (5.22). Here we consider the}

elements x = C1, y = C2, and C3, . . . , C8 as elements of N4(x, y). The images of the

elements C1, . . . , C8 by the natural epimorphism N4(x, y) → N4(x, y) /R we also denote

by C1, . . . , C8. We see from Proposition 5.2 that the base of the group N4(x, y) /R is

{C1, C2, . . . , C7} and, if 1 ≤ i ≤ 3, then |Ci| = 4, if 4 ≤ i ≤ 7, then |Ci| = 2.

Our goal is to prove that N4(x, y) /R = FΘ(x, y). It means that for computation

of the two generated free group of variety Θ enough to factorize the two generated free group of variety N4 by the normal group of relations which we already conclude from the

identities of the variey Θ. For this we must study the group N4(x, y) /Rand prove some

lemmas about it's properties. These lemmas we will use in the proof of Theorem 7.1 and in the other computations.

6 Some lemmas about the group

N

₄

(x, y) /R

In this section we will denote the group N4(x, y) /R by G.

Lemma 6.1 γ2(G) is a commutative group.

Proof. We have that γ3(N4(x, y)) ≤ Z (γ2(N4(x, y))) and quotient group

γ2(N4(x, y)) /γ3(N4(x, y)) = h(y, x) γ3(N4(x, y))iis a cyclic group. Therefore γ2(N4(x, y))

is a commutative group. G is a homomorphic image of N4(x, y), so, γ2(G) is a

commu-tative group.

Lemma 6.2 The group γ3(G) is a group of exponent 2.

Proof. We have that γ3(G) = hC4, . . . , C7i. Lemma 6.1 and the consideration of relations

(5.21) they complete the proof.

Lemma 6.3 For every h ∈ γ2(G) the inclusion h2 ∈ γ4(G) holds.

Proof. We have that γ2(G) = hC3, . . . , C7i. Lemma 6.1 and the consideration of relations

(5.21) and (5.22) complete the proof.

Lemma 6.4 For every a, b, c ∈ G the following equalities holds:

(ab, c)2 = (a, c)2(b, c)2, (6.1)

(a, bc)2 = (a, c)2(a, b)2 (6.2) (a−1, b)2 = (a, b)2 (6.3) (a, b−1)2 = (a, b)2. (6.4)

Proof. We have that

(ab, c)2 = (a, c)b(b, c)
2 = (a, c)2
b(b, c)2

by (5.1) and by Lemma 6.1. And now by Lemma 6.3 we conclude (6.1). By similar com-putation we can conclude (6.1) from (5.2) and Lemma 6.3.

By Lemmas 6.3 and 6.2 γ2(G) is a group of exponent 4. Therefore, by (5.3) and

Lemma 6.3 we have that

(a−1, b)2 =(b, a)a−12 = (b, a)2
a−1

= (b, a)2 = (a, b)−2 = (a, b)2. By similar computation we can conclude (6.4).

Lemma 6.5 If g ∈ G, h ∈ γ2(G), then (gh) 4

= g4.

Proof. We know that the identity (5.11) holds in the variety N4. So this identity holds

in G. Hence we have that

(gh)4 = g4h4(h, g)6(h, g, h)14(h, g, h, h)11(h, g, g)4(h, g, g, h)11(h, g, g, g).

In our case (h, g, h), (h, g, h, h), (h, g, g, h), (h, g, g, g) ∈ γ5(G). By Lemmas 6.3 and 6.2 we

7 Computation of the group

F

_Θ

(x, y)

Theorem 7.1 N4(x, y) /R = FΘ(x, y).

Proof. In this proof we also denote the group N4(x, y) /R by G.

By Proposition 5.2 the relations r = 1, where r ∈ R, are consequences from the identities which dene the variety Θ. So we only must prove that G ∈ Θ.

It is clear that the group G is a nilpotent groups of class 4. As we said in the proof of Lemma 6.3, G is a metabelian group.

Now we will prove that the group G fullls identity (2.6). By Lemma 6.5, it remains for us to prove now that for every 0 ≤ α1, α2 ≤ 3

(xα1_yα2₎4 _{= 1}

holds in G. We substitute in (5.11) xα1 instead x and yα2 instead y. (xα1₎4 _{= (y}α2₎4 _{= 1}

holds in G. Therefore we must only prove that

(yα2_{, x}α1₎6_(yα2_{, x}α1_{, y}α2₎14_(yα2_{, x}α1_{, y}α2_{, y}α2₎11_(yα2_{, x}α1_{, x}α1₎4_·

(yα2_{, x}α1_{, x}α1_{, y}α2₎11_(yα2_{, x}α1_{, x}α1_{, x}α1_{) = 1}

holds in G. By Lemmas 6.3 and 6.2 we have that

(yα2_{, x}α1₎6 _{= (y}α2_{, x}α1₎2_,

(yα2_{, x}α1_{, y}α2₎14_{= 1,}

(yα2_{, x}α1_{, y}α2_{, y}α2₎11_{= (y}α2_{, x}α1_{, y}α2_{, y}α2_),

(yα2_{, x}α1_{, x}α1₎4 _{= 1,}

We denote

v (α1, α2) = (yα2, xα1)2(yα2, xα1, yα2, yα2)(yα2, xα1, xα1, yα2)(yα2, xα1, xα1, xα1)

So it remains for us to prove that

v (α1, α2) = 1 (7.1)

holds in G for every 0 ≤ α1, α2 ≤ 3.

It is clear that (7.1) holds in G if α1 = 0 or α2 = 0. If α1 = α2 = 1, than by (5.11),

we have that

v (1, 1) = (y, x)2(y, x, y, y)(y, x, x, y)(y, x, x, x) = C₃2C7C8C6 = 1.

Now we will prove (7.1) by induction on α1 and α2. We suppose that (7.1) holds for all

α1, α2 such that α1 ≤ β1, α2 ≤ β2 and 0 ≤ α1, α2. We have by (6.2) that

(yβ2_{, x}β1+1₎2 _{= y}β2_{, x}
2 _yβ2_{, x}β1
2_. (7.2) We have by (5.8) that (yβ2_{, x}β1+1_{, y}β2_{, y}β2_{) = (y}β2_{, x}β1_{, y}β2_{, y}β2_)(yβ2_{, x, y}β2_{, y}β2_), (7.3) (yβ2_{, x}β1+1_{, x}β1+1_{, y}β2_{) = (y}β2_{, x, x, y}β2₎(β1+1)2 ₌ (yβ2_{, x, x, y}β2₎β12_(yβ2_{, x, x, y}β2₎2β1_(yβ2_{, x, x, y}β2₎ and (yβ2_{, x}β1+1_{, x}β1+1_{, x}β1+1_{) = (y}β2_{, x, x, x)}(β1+1)3 ₌ (yβ2_{, x, x, x)}β31_(yβ2_{, x, x, x)}3β1(β1+1)_(yβ2_{, x, x, x).}

By Lemma 6.2 we have that

(yβ2_{, x, x, y}β2₎2β1 _{= (y}β2_{, x, x, x)}3β1(β1+1) _{= 1,}

because β1(β1+ 1) is an even number. Hence

(yβ2_{, x}β1+1_{, x}β1+1_{, y}β2_{) = (y}β2_{, x, x, y}β2₎β12(yβ2_{, x, x, y}β2_{) =} (7.4)

(yβ2_{, x}β1_{, x}β1_{, y}β2_)(yβ2_{, x, x, y}β2₎

and

(yβ2_{, x}β1_{, x}β1_{, x}β1_)(yβ2_{, x, x, x).}

Therefore, by (7.2), (7.3), (7.4), (7.5) and by our hypothesis about v (β1, β2)and v (1, β2),

we have that

v (β1+ 1, β2) = v (β1, β2) v (1, β2) = 1.

By (6.1) we have that

yβ2+1_{, x}β1
2 _{= y}β2_{, x}β1
2 _{y, x}β1
2_, (7.6)

And now, similarly to the previous arguments, we conclude that

(yβ2+1_{, x}β1_{, x}β1_{, x}β1_{) = (y}β2_{, x}β1_{, x}β1_{, x}β1_{)(y, x}β1_{, x}β1_{, x}β1_), (7.7)

(yβ2+1_{, x}β1_{, x}β1_{, y}β2+1_{) = (y}β2_{, x}β1_{, x}β1_{, y}β2_{)(y, x}β1_{, x}β1_{, y),} (7.8)

and

(yβ2+1_{, x}β1_{, y}β2+1_{, y}β2+1_{) = (y}β2_{, x}β1_{, y}β2_{, y}β2_{)(y, x}β1_{, y, y).} (7.9)

Hence, by (7.6), (7.7), (7.8), (7.9) and by our hypothesis about v (β1, β2)and v (β1, 1),

v (β1, β2+ 1) = v (β1, β2) v (β1, 1) = 1.

Therefore we prove that (7.1) holds in G for every 0 ≤ α1, α2 ≤ 3. This completes our

proof.

Now, when we know that N4(x, y) /R = FΘ(x, y), we can prove

Corollary 1 The lemmas 6.1, 6.2, 6.3 and 6.4 hold when we consider as group G an arbitrary group of the variety Θ.

Proof. Lemma 6.1 is fullled by denition of the variety Θ.

Now we will prove that every G ∈ Θ fullls the conclusion of Lemma 6.3. Every h ∈ γ2(G) is generated by commutators (a, b), such that a, b ∈ G. There exists the

homomorphism ϕ : FΘ(x, y) → G such that ϕ (x) = b, ϕ (y) = a. We apply ϕ to (5.22)

and conclude that (a, b)2

∈ γ4(G).

Also every G ∈ Θ satises the conclusion of Lemma 6.2, because the group γ3(G)

is generated by the commutators (a, b), where a ∈ G, b ∈ γ2(G). We also consider the

homomorphism ϕ : FΘ(x, y) → G from the previous part of the proof, apply it to (5.22)

and now, because b ∈ γ2(G), conclude that (a, b)2 ∈ γ5(G).

8 Applicable systems of words.

Necessary conditions

Proposition 8.1 If W (see (4.1) ) is an applicable system of words in our variety Θ then always w1 = 1, w−1(x) = x−1.

Proof. We suppose that W is an applicable system of words. w1 ∈ FΘ(∅) = {1}, so

w1 = 1.

w−1(x) ∈ FΘ(x) ∼= Z4. We denote FΘ(x)by F . Because W is an applicable system of

words, by Denition 3.5, there exists the isomorphism sF : F → FW∗ , such that sF (x) = x.

We have that sF(x−1) = w−1(sF (x)) = w−1(x). If w−1(x) = 1, then sF (x−1) = 1, but

sF(1) = w1 = 1, but it contradicts with the assumption that sF is an injective mapping.

If w−1(x) = x, then sF(x−1) = x = sF (x)which gives the same contradiction.

If w−1(x) = x2, then x = sF (x) = sF



(x−1)−1 = w−1(w−1(x)) = w−1(x2) =

(x2)2 = x4 = 1. It also gives us a contradiction.

Therefore, there is only one possibility: w−1(x) = x3 = x−1.

For the studying of words w·(x, y)we need to consider the group FΘ(x, y). We denote

this group by G.

Because W is an applicable system of words, by Denition 3.5, there exists the iso-morphism sG: G → G∗W, which x x and y.

Proposition 8.2 If W (see (4.1) ) is an applicable system of words in our variety Θ then always w·(x, y) = xyC3α3C α4 4 C α5 5 C α6 6 C α7 7 , (8.1) where 0 ≤ α3 < 4, αi ∈ {0, 1}, when 4 ≤ i ≤ 7.

Proof. We use the considerations of (TSURKOV, 2007b, Proposition 2.1). w·(x, y) ∈ G,

We have that x = sG(x · 1) = w·(sG(x) , sG(1)) = w·(x, w1) = w·(x, 1) = xα1g2(x, 1) =

xα1 holds, because g

2(x, 1)is the result of substitution of 1 instead y in g2(x, y). Therefore,

α1 = 1. We obtain by similar computations that α2 = 1.

In the next Proposition we will get the stronger result about the word w·(x, y)from

applicable system of words W .

Proposition 8.3 If W (see (4.1) ) is an applicable system of words in our variety Θ then always w·(x, y) = xyC3α3, where α3 = 0, 1, 2, 3.

Proof. The equalities x (xy) = (xx) y and x (yy) = (xy) y hold in G = FΘ(x, y). We

apply the isomorphism sG : G → G∗W to the both hands of the rst equality and we have

that sG(x (xy)) = w·(sG(x) , sG(xy)) = w·(sG(x) , w·(sG(x) , sG(y))) = w·(x, w·(x, y)) and sG((xx) y) = w·(sG(xx) , sG(y)) = w·(w·(sG(x) , sG(x)) , sG(y)) = w·(w·(x, x) , y) . Therefore w·(x, w·(x, y)) = w·(w·(x, x) , y) .

We conclude by the similar computations from the second equality that w·(x, w·(y, y)) = w·(w·(x, y) , y)

holds. If we will denote the operation dened by the word (8.1) by symbol ◦, then we can rewrite these equalities in the form

x ◦ (x ◦ y) = (x ◦ x) ◦ y (8.2) and

x ◦ (y ◦ y) = (x ◦ y) ◦ y. (8.3) Now we will compute the left hand of (8.2). We have that

x ◦ (x ◦ y) = x ◦ xyCα3 3 C α4 4 C α5 5 C α6 6 C α7 7 = xxyCα3 3 C α4 4 C α5 5 C α6 6 C α7 7 · L α3 3 L α4 4 L α5 5 L α6 6 L α7 7 , (8.4)