On a decomposition of an element of a free metabelian group as a productof primitive elements

СОДЕРЖАНИЕ: Доказано, что произвольный элемент свободной метабелевой группы представимкак произведение не более четырех примитивных элементов. Таким образом,установлено,что примитивная ширина свободной метабелевой группы конечна.

E.G. Smirnova, Omsk State University, Mathematical Department

1. Introduction

Let G=Fn/V be a free in some variety group of rank n. An element

On a decomposition of an element of a free metabelian group as a productof primitive elementsis called primitive if and only if g can be included in some basis g=g1,g2,...,gn of G. The aim of this note is to consider a presentation of elements of free groups in abelian and metabelian varieties as a product of primitive elements. A primitive length |g|pr of an element
On a decomposition of an element of a free metabelian group as a productof primitive elementsis by definition a smallest number m such that g can be presented as a product of m primitive elements. A primitive length |G|pr of a group G is defined as
On a decomposition of an element of a free metabelian group as a productof primitive elements, so one can say about finite or infinite primitive length of given relatively free group.

Note that |g|pr is invariant under action of Aut G. Thus this notion can be useful for solving of the automorphism problem for G.

This note was written under guideness of professor V. A. Roman'kov. It was supported by RFFI grant 95-01-00513.

2. Presentation of elements of a free abelian group of rank n as a product of primitive elements

Let An be a free abelian group of rank n with a basis a1,a2,...,an. Any element

On a decomposition of an element of a free metabelian group as a productof primitive elementscan be uniquelly written in the form

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Every such element is in one to one correspondence with a vector

On a decomposition of an element of a free metabelian group as a productof primitive elements. Recall that a vector (k1,...,kn) is called unimodular, if g.c.m.(k1,...,kn)=1.

Лемма 1. An element

On a decomposition of an element of a free metabelian group as a productof primitive elementsof a free abelian group An is primitive if and only if the vector (k1,...,kn) is unimodular.

Доказательство. Let

On a decomposition of an element of a free metabelian group as a productof primitive elements, then
On a decomposition of an element of a free metabelian group as a productof primitive elements. If c is primitive, then it can be included into a basis c=c1,c2,...,cn of the group An. The group
On a decomposition of an element of a free metabelian group as a productof primitive elements(n factors) in such case, has a basis
On a decomposition of an element of a free metabelian group as a productof primitive elements, where
On a decomposition of an element of a free metabelian group as a productof primitive elementsmeans the image of ci. However,
On a decomposition of an element of a free metabelian group as a productof primitive elements, that contradics to the well-known fact: An(d) is not allowed
On a decomposition of an element of a free metabelian group as a productof primitive elementsgenerating elements. Conversely, it is well-known , that every element c=a1k1,...,ankn such that g.c.m.(k1,...,kn)=1 can be included into some basis of a group An.

Note that every non unimodular vector

On a decomposition of an element of a free metabelian group as a productof primitive elementscan be presented as a sum of two unimodular vectors. One of such possibilities is given by formula (k1,...,kn)=(k1-1,1,k3,...,kn)+(1,k2-1,0,...,0).

Предложение 1. Every element

On a decomposition of an element of a free metabelian group as a productof primitive elements,
On a decomposition of an element of a free metabelian group as a productof primitive elements, can be presented as a product of not more then two primitive elements.

Доказательсво. Let c=a1k1...ankn for some basis a1,...an of An. If g.c.m.(k1,...,kn)=1, then c is primitive by Lemma 1. If

On a decomposition of an element of a free metabelian group as a productof primitive elements, then we have the decomposition (k1,...,kn)=(s1,...,sn)+(t1,...,tn) of two unimodular vectors. Then c=(a1s1...ansn)(a1t1...antn) is a product of two primitive elements.

Corollary.It follows that |An|pr=2 for

On a decomposition of an element of a free metabelian group as a productof primitive elements. ( Note that
On a decomposition of an element of a free metabelian group as a productof primitive elements.

3. Decomposition of elements of the derived subgroup of a free metabelian group of rank 2 as a product of primitive ones

Let

On a decomposition of an element of a free metabelian group as a productof primitive elementsbe a free metabelian group of rank 2. The derived subgroup M'2 is abelian normal subgroup in M2. The group
On a decomposition of an element of a free metabelian group as a productof primitive elementsis a free abelian group of rank 2. The derived subgroup M'2 can be considered as a module over the ring of Laurent polynomials

On a decomposition of an element of a free metabelian group as a productof primitive elements.

The action in the module M'2 is determined as

On a decomposition of an element of a free metabelian group as a productof primitive elements,where
On a decomposition of an element of a free metabelian group as a productof primitive elementsis any preimage of element
On a decomposition of an element of a free metabelian group as a productof primitive elementsin M2, and

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Note that for

On a decomposition of an element of a free metabelian group as a productof primitive elements,
On a decomposition of an element of a free metabelian group as a productof primitive elementswe have

(u,g)=ugu-1g-1=u1-g.

Any automorphism

On a decomposition of an element of a free metabelian group as a productof primitive elementsis uniquelly determined by a map

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Since M'2 is a characteristic subgroup,

On a decomposition of an element of a free metabelian group as a productof primitive elementsinduces automorphism
On a decomposition of an element of a free metabelian group as a productof primitive elementsof the group A2 such that

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements

Consider an automorphism

On a decomposition of an element of a free metabelian group as a productof primitive elementsof the group M2, identical modM'2, which is defined by a map

On a decomposition of an element of a free metabelian group as a productof primitive elements,

On a decomposition of an element of a free metabelian group as a productof primitive elements

By a Bachmuth's theorem from [1]

On a decomposition of an element of a free metabelian group as a productof primitive elementsis inner, thus for some
On a decomposition of an element of a free metabelian group as a productof primitive elementswe have

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements

Consider a primitive element of the form ux,

On a decomposition of an element of a free metabelian group as a productof primitive elements. By the definition there exists an automorphism
On a decomposition of an element of a free metabelian group as a productof primitive elementssuch that

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements
(1)

On a decomposition of an element of a free metabelian group as a productof primitive elements

Using elementary transformations we can find a IA-automorphism with a first row of the form(1). Then by mentioned above Bachmuth's theorem

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements

In particular the elements of type u1-xx, u1-yy,

On a decomposition of an element of a free metabelian group as a productof primitive elementsare primitive.

Предложение 2. Every element of the derived subgroup of a free metabelian group M2 can be presented as a product of not more then three primitive elements.

Доказательство. Every element

On a decomposition of an element of a free metabelian group as a productof primitive elementscan be written as
On a decomposition of an element of a free metabelian group as a productof primitive elements, and
On a decomposition of an element of a free metabelian group as a productof primitive elementscan be presented as

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements.

Thus,

On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements
(2)

A commutator

On a decomposition of an element of a free metabelian group as a productof primitive elements, by well-known commutator identities can be presented as

On a decomposition of an element of a free metabelian group as a productof primitive elements
(3)

The last commutator in (3) can be added to first one in (2). We get

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements[y-1
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements, that is a product of three primitive elements.

4. A decomposition of an element of a free metabelian group of rank 2 as a product of primitive elements

For further reasonings we need the following fact: any primitive element

On a decomposition of an element of a free metabelian group as a productof primitive elementsof a group A2 is induced by a primitive element
On a decomposition of an element of a free metabelian group as a productof primitive elements,
On a decomposition of an element of a free metabelian group as a productof primitive elements. It can be explained in such way. One can go from the basis
On a decomposition of an element of a free metabelian group as a productof primitive elementsto some other basis by using a sequence of elementary transformations, which are in accordance with elementary transformations of the basis <x,y> of the group M2.

The similar assertions are valid for any rank

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Предложение 3. Any element of group M2 can be presented as a product of not more then four primitive elements.

Доказательство. At first consider the elements in form

On a decomposition of an element of a free metabelian group as a productof primitive elements. An element
On a decomposition of an element of a free metabelian group as a productof primitive elementsis primitive in A2 by lemma 1, consequently there is a primitive element of type
On a decomposition of an element of a free metabelian group as a productof primitive elements. Hence,
On a decomposition of an element of a free metabelian group as a productof primitive elementsSince, an element
On a decomposition of an element of a free metabelian group as a productof primitive elementsis primitive, it can be included into some basis
On a decomposition of an element of a free metabelian group as a productof primitive elementsinducing the same basis
On a decomposition of an element of a free metabelian group as a productof primitive elementsof A2. After rewriting in this new basis we have:

On a decomposition of an element of a free metabelian group as a productof primitive elements,

and so as before

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements

Obviously, two first elements above are primitive. Denote them as p1, p2. Finally, we have

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements, a product of three primitive elements.

If

On a decomposition of an element of a free metabelian group as a productof primitive elements, then by proposition 1 we can find an expansion
On a decomposition of an element of a free metabelian group as a productof primitive elementsas a product of two primitive elements, which correspond to primitive elements of M2: v1xk1yl1,v2xk2yl2,v1,v2
On a decomposition of an element of a free metabelian group as a productof primitive elements.

Further we have the expansion

On a decomposition of an element of a free metabelian group as a productof primitive elements

The element w(v1xk1yl1) can be presented as a product of not more then three primitive elements. We have a product of not more then four primitive elements in the general case.

5. A decomposition of elements of a free metabelian group of rank

On a decomposition of an element of a free metabelian group as a productof primitive elementsas a product of primitive elements

Consider a free metabelian group Mn=<x1,...,xn> of rank

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Предложение 4. Any element

On a decomposition of an element of a free metabelian group as a productof primitive elementscan be presented as a product of not more then four primitive elements.

Доказательсво. It is well-known [2], that M'n as a module is generated by all commutators

On a decomposition of an element of a free metabelian group as a productof primitive elements. Therefore, for any
On a decomposition of an element of a free metabelian group as a productof primitive elementsthere exists a presentation

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements

Separate the commutators from (4) into three groups in the next way.

1)

On a decomposition of an element of a free metabelian group as a productof primitive elements- the commutators not including the element x2 but including x1.

2)

On a decomposition of an element of a free metabelian group as a productof primitive elements - the other commutators not including the x1.

3) And the third set consists of the commutator

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Consider an automorphism of Mn, defining by the following map:

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements,

On a decomposition of an element of a free metabelian group as a productof primitive elements.

The map

On a decomposition of an element of a free metabelian group as a productof primitive elementsdetermines automorphism, since the Jacobian has a form

On a decomposition of an element of a free metabelian group as a productof primitive elements,

and hence, det Jk=1.

Since element

On a decomposition of an element of a free metabelian group as a productof primitive elementscan be included into a basis of Mn, it is primitive. Thus any element
On a decomposition of an element of a free metabelian group as a productof primitive elementscan be presented in form
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements

On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elementsx3x2x1]

[x1-1x2-1x3-1]. =p1p2p3p4 a product of four primitive elements.

Note that the last primitive element p4=x1-1x2-1x3-1 can be arbitrary.

Предложение 5. Any element of a free metabelian group Mn can be presented as a product of not more then four primitive elements.

Доказательство. Case 1. Consider an element

On a decomposition of an element of a free metabelian group as a productof primitive elements, so that g.c.m.(k1,...,kn)=1. An element
On a decomposition of an element of a free metabelian group as a productof primitive elementsis primitive by lemma 1 and there exists a primitive element
On a decomposition of an element of a free metabelian group as a productof primitive elements,
On a decomposition of an element of a free metabelian group as a productof primitive elements

An element from derived subgroup can be presented as a product of not more then four primitive elements with a fixed one of them:

On a decomposition of an element of a free metabelian group as a productof primitive elements

Then

On a decomposition of an element of a free metabelian group as a productof primitive elements.

Case 2. If

On a decomposition of an element of a free metabelian group as a productof primitive elements, then by lemma 2
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements, where
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elementsare primitive in An. There exist primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elements
On a decomposition of an element of a free metabelian group as a productof primitive elementsSo
On a decomposition of an element of a free metabelian group as a productof primitive elementsWe have just proved that the element wp1 can be presented as a product of not more then three primitive elements p1'p2'p3'. Finally we have c=p1'p2'p3'p2, a product of not more then four primitive elements.

Список литературы

Bachmuth S. Automorphisms of free metabelian groups // Trans.Amer.Math.Soc. 1965. V.118. P. 93-104.

Линдон Р., Шупп П. Комбинаторная теория групп. М.: Мир, 1980.

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