_{1}

^{*}

The paper considers the problem of full transitivity of a cotorsion hull of a separable primary group **G** when a ring of endomorphisms **E**(**G**) of the group **G** has the form , where ** E_{s(G)}** is a subring of small endomorphisms of the ring

**E**(

**G**), whereas

**J**

_{p}is a ring of integer

**P**-adic numbers. Investigation of the issue of full transitivity of a group is essentially helpful in studying its fully invariant subgroups as well as the lattice formed by these subgroups. It is proved that in the considered case, the cotorsion hull is not fully transitive. A lemma is proposed, which can be used in the study of full transitivity of a group and in other cases.

The groups discussed in the paper are abelian and the operation is written in additive terms. We use here the notation and terminology of the monographs [

The symbol

The knowledge of the construction of fully invariant subgroups of an abelian group and their lattice is essentially helpful in the study of the properties of the group itself and also in the investigation of the properties of its rings of endomorphisms and quasi-endomorphisms, the group of automorphisms and other algebraic systems connected with the initial group.

For a sufficiently wide class of

However little is known about the results obtained in this area for the class of cotorsion groups. A group

where

It is noteworthy that endomorpohisms in cotorsion groups are completely defined by their action on the torsion part and, as shown by W. May and E. Toubassi [

The notion of full transitivity of a group plays an essential role in describing the lattice of fully invariant subgroups.

By the

where

A reduced

Using the indicators of fully transitive groups we can describe the lattice of fully invariant subgroups (see [

For a module over a commutative ring, A. Mader formulated a general scheme that can be used to describe the lattice of fully invariant submodules of the module (see [

In the same way as we did for a

R. Pierce [

where

For all

The Pierce group

As mentioned above, R. Pierce [

To study the full transitivity of the group

Representation of elements in this form makes it easy to calculate the height and the indicator. In particular, if

where

Let

The following lemma is true.

Lemma 2.1. If

Proof. Consider two elements

of the group

be the element of the group

where

The commutativity of diagram (2.4) immediately follows from the definition of these homomorphisms.

To extension (2.3) there corresponds the sequence

Then for each

For an endomorphism

It is obvious that the right-hand part of equality (2.5) defines the extension of an endomorphism

Now we can consider an element

of the group

Let

where

where

Thus we have shown that (2.4) and (2.8) are commutative diagrams. Then, according to ([

For the Pierce group

Theorem 2.1. The cotorsion hull

Proof. We use representation (2.1) of cotorsion hull elements and assume that

where

By (2.2) we have

Let

Since

and

Therefore

On the other hand, from (2.10) we obtain

Therefore

But

Thus there exists no endomorphism

Note that one more example of a separable primary group, the cotorsion hull of which is not fully transitive, can be found in ([

As mentioned above, if the separable primary group

This study was supported by the grant (ATSU-2013/44) of Akaki Tsereteli University.