On the Full Transitivity of a Cotorsion Hull of the Pierce Group ()
1. Introduction
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 [1] [2] .
The symbol 
denotes a fixed prime number. 
and 
are respectively the groups of integer and rational numbers. A subgroup 
of the group 
is called fully invariant if it is self-mapped for any endomorphism of the group
.
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 
-groups these topics were studied by R. Baer, I. Kaplansky, P. Linton, R. Pierce, D. Moore, E. Hewett and others. The works of A. Mader, R. Göbel, P. A. Krylov, S. Ya. Grinshpon, A. I. Moskalenko and other authors are dedicated to the investigation of these topics in torsion-free and mixed groups.
However little is known about the results obtained in this area for the class of cotorsion groups. A group 
is called a cotorsion group if its extension by means of any torsion-free group 
splits as follows:
. The importance of the class of cotorsion groups in the theory of abelian groups is due to two factors: for any groups
, 
, the group 
is a cotorsion one and any reduced group 
is isomorphically embeddable in the group 
called the cotorsion hull of the group
. If the torsion part of the group 
is denoted by
, then
where
. Thus the study of cotorsion groups essentially reduces to the study of groups of the form
, where 
is a 
-primary group.
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 [3] , for a mixed group 
the ring of endomorphisms 
is isomorphic to 
if and only if 
is a fully invariant subgroup of the cotorsion hull
.
The notion of full transitivity of a group plays an essential role in describing the lattice of fully invariant subgroups.
By the 
-indicator of an element 
of the group 
we mean an increasing sequence of ordinal numbers
where 
is the generalized 
-height of an element, i.e. for 
if 
and
. Now for the set of indicators we can introduce the order
A reduced 
-group is called fully transitive if for arbitrary elements 
and
, when 
there exists an endomorphism 
of the group such that
. The class of fully transitive groups includes such important groups as separable 
-groups, algebraically compact groups and quasi-pure injective groups.
Using the indicators of fully transitive groups we can describe the lattice of fully invariant subgroups (see [4] -[11] ).
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 [10] , Theorem 2.1 or [12] , Theorem 1.1).
In the same way as we did for a 
-group we define the notion of full transitivity for the group
. According to A. Mader [10] , an algebraically compact group is fully transitive and described with the aid of indicators the lattice of fully invariant subgroups of this group. This means to describe the lattice of fully invariant subgroups of the group 
when 
is a torsion-complete group. When 
is the direct sum of cyclic 
-groups, A. Moskalenko [11] proved that 
is also fully transitive and described by means of indicators the lattice of fully invariant subgroups of the group
. In general, for the separable primary group
, the cotorsion hull 
is not fully transitive. In particular if 
is an infinite direct sum of torsion-complete groups, then, as shown by the author [13] , the group 
is not fully transitive and in that case the lattice of fully invariant subgroups of the group 
cannot be described by means of indicators (see [12] ).
R. Pierce [14] considered the primary group
, a ring of whose endomorphisms has the form
(1.1)
where 
is the ring of small endomorphisms of the group 
which is the ideal of the ring of endomorphisms 
of the group
, whereas 
is the ring of integer 
-adic numbers. A small endomorphism of the group 
is defined as follows (see [14] ).
For all 
there exists an integer 
such that
. (1.2)
The Pierce group 
is important when studying the ring of endomorphisms of abelian groups (see [15] ). The aim of the present paper consists in elucidating the full transitivity of the cotorsion hull 
and also in finding the conditions, under which the cotorsion hull is not fully transitive.
2. Full Transitivity of the Cotorsion Hull of the Pierce Group
As mentioned above, R. Pierce [14] considered the separable primary group 
with a standard basic subgroup
, 
, 
, 
, where 
is a torsion-complete group, i.e. the torsion part of a 
-adic completion of the group
. The cardinality is 
and the ring of endomorphisms of the group 
has form (1.1).
To study the full transitivity of the group
, we use the following representation of elements of the cotorsion hull of 
given by A. Moskalenko [11] for the separable 
-group 
. (2.1)
Representation of elements in this form makes it easy to calculate the height and the indicator. In particular, if
, then
(2.2)
where 
is the smallest infinite ordinal number.
Let 
be a basic subgroup of the reduced separable 
-group 
lying between 
and
. Elements
, 
,
. As is know, an endomorphism 
of the group 
extends uniquely to an endomorphism of
.
The following lemma is true.
Lemma 2.1. If 
and there exists no endomorphism 
of the group 
for which
, then a cotorsion hull 
is not fully transitive.
Proof. Consider two elements
of the group
. Then by the condition of the theorem and (2.2) we have
. As is known, each endomorphism of the group 
extends uniquely to an endomorphism of the group
. We will show that if for an endomorphism
, 
, then
. Let
(2.3)
be the element of the group 
defined by the sequence
. For an endomorphism 
of the group 
let us show that
. According to ([1] , Section 50), the extension of 
is defined from the commutative diagram
(2.4)
where 
is the identical inclusion,
The commutativity of diagram (2.4) immediately follows from the definition of these homomorphisms.
To extension (2.3) there corresponds the sequence
, where elements are defined as follows: fix a system of generators 
of the group
, 
,
. Let
, 
, be a system of representatives of the adjacent classes 
of the group
, 
, 
,
. Denote
.
Then for each
,
.
For an endomorphism 
of the group 
we have
, 
, and can define
(2.5)
It is obvious that the right-hand part of equality (2.5) defines the extension of an endomorphism 
on 
and if 
is some other endomorphism of the group
, which induces 
on
, then 
contains 
and 
([1] , Proposition 34.1). From (2.5) we have
.
Now we can consider an element
(2.6)
of the group 
and with its aid define the corresponding short exact sequence.
Let 
be the group defined by a system of generators 
which are defined by the relations of the group 
and the equalities
, 
,
. Then
(2.7)
where 
is the identical inclusion and, for each element
, 
, 
,
is a short exact sequence. To extension (2.7) there corresponds sequence (2.6) (see [11] , Proof of Theorem 1). Let us show that by using extensions (2.3) and (2.7) we can compose the commutative diagram
(2.8)
where 
is the above-mentioned endomorphism and
, 
, 
,
. Indeed, from the definition of a triple 
we immediately conclude that (2.8) is a commutative diagram.
Thus we have shown that (2.4) and (2.8) are commutative diagrams. Then, according to ([1] , Section 50), 
and 
are equivalent extensions and thereby define one and the same sequence from
. But, by virtue of our construction, 
is the sequence corresponding to the extension
; therefore it corresponds to the extension
, too. Thus
. Therefore if the endomorphism 
maps the element 
into
, then
, i.e. we have proved more than what has been mentioned at the beginning of the proof of the lemma. Thus it obviously follows that if there exists no endomorphism 
of the group 
for which
, then there exists no endomorphism 
of the group 
which maps the element 
into
, i.e. 
is not fully transitive. The lemma is proved.
For the Pierce group 
the following statement is true.
Theorem 2.1. The cotorsion hull 
of the group 
is not fully transitive.
Proof. We use representation (2.1) of cotorsion hull elements and assume that 
and 
are elements of the group
, where
,
. By virtue of (11, Item 2), elements 
and 
can be written in the form
where
,
. Since 
and 
is infinite, taking into account ([14] : Lemma 15.1, Theorem 15.4) we can assume that
(2.9)
By (2.2) we have
, i.e. the following condition is fulfilled
.
Let 
be an endomorphism of the group
. Using (1.1) we have
, where 
is a small endomorphism of the group 
and 
is the 
-adic number
. As is known ([1] , Section 39), the endomorphism 
uniquely extends to the endomorphism 
of the group 
Since
(2.10)
and 
is a small endomorphism of the group 
(see (1.2)), starting with some 
we have
.
Therefore
On the other hand, from (2.10) we obtain
Therefore
But 
since
, 
, and in that case the equality 
would contradict condition (2.9). Therefore
.
Thus there exists no endomorphism 
of the group 
which extends to the endomorphism 
of the group 
and
. Then from Lemma 1.1 it follows that Theorem 2.1 is valid.
Note that one more example of a separable primary group, the cotorsion hull of which is not fully transitive, can be found in ([11] , item 3).
As mentioned above, if the separable primary group 
is a direct sum of cyclic 
-groups or a cotorsioncomplete group, then the cotorsion hull 
is fully transitive. In 1993, at Professor A. Fomin’s seminar A. Moskalenko made a conjecture that 
is fully transitive only in these two cases. The proved lemma and theorem may serve as a positive argument in favor of this conjecture.
Acknowledgements
This study was supported by the grant (ATSU-2013/44) of Akaki Tsereteli University.