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

Consider a primitive element of the form ux, . By the definition there exists an automorphism such that

(1)

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

In particular the elements of type u1-xx, u1-yy, are 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 can be written as , and can be presented as

.

Thus,

(2)

A commutator , by well-known commutator identities can be presented as

(3)

The last commutator in (3) can be added to first one in (2). We get [y-1 , 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 of a group A2 is induced by a primitive element , . It can be explained in such way. One can go from the basis to 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 .

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 . An element is primitive in A2 by lemma 1, consequently there is a primitive element of type . Hence, Since, an element is primitive, it can be included into some basis inducing the same basis of A2. After rewriting in this new basis we have:

,

and so as before

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

, a product of three primitive elements.

If , then by proposition 1 we can find an expansion as a product of two primitive elements, which correspond to primitive elements of M2: v1xk1yl1,v2xk2yl2,v1,v2 .

Further we have the expansion

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 as a product of primitive elements